TOPICS
SURROUNDING
THE
COMBINATORIAL
ANABELIAN
GEOMETRY
OF
HYPERBOLIC
CURVES
III:
TRIPODS
AND
TEMPERED
FUNDAMENTAL
GROUPS
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
JUNE
2023
Abstract.
Let
Σ
be
a
subset
of
the
set
of
prime
numbers
which
is
either
equal
to
the
entire
set
of
prime
numbers
or
of
cardinal-
ity
one.
In
the
present
paper,
we
continue
our
study
of
the
pro-Σ
fundamental
groups
of
hyperbolic
curves
and
their
associated
con-
figuration
spaces
over
algebraically
closed
fields
in
which
the
primes
of
Σ
are
invertible.
The
focus
of
the
present
paper
is
on
appli-
cations
of
the
theory
developed
in
previous
papers
to
the
theory
of
tempered
fundamental
groups,
in
the
style
of
André.
These
applications
are
motivated
by
the
goal
of
surmounting
two
funda-
mental
technical
difficulties
that
appear
in
previous
work
of
André,
namely:
(a)
the
fact
that
the
characterization
of
the
local
Galois
groups
in
the
global
Galois
image
associated
to
a
hyperbolic
curve
that
is
given
in
earlier
work
of
André
is
only
proven
for
a
quite
limited
class
of
hyperbolic
curves,
i.e.,
a
class
that
is
“far
from
generic”;
(b)
the
proof
given
in
earlier
work
of
André
of
a
certain
key
injectivity
result,
which
is
of
central
importance
in
establishing
the
theory
of
a
“p-adic
local
analogue”
of
the
well-known
“global”
theory
of
the
Grothendieck-Teichmüller
group,
contains
a
fundamental
gap.
In
the
present
paper,
we
surmount
these
technical
difficulties
by
introduc-
ing
the
notion
of
an
“M-admissible”,
or
“metric-admissible”,
outer
automorphism
of
the
profinite
geometric
fundamental
group
of
a
p-adic
hyperbolic
curve.
Roughly
speaking,
M-admissible
outer
automorphisms
are
outer
automorphisms
that
are
compatible
with
the
data
constituted
by
the
indices
at
the
various
nodes
of
the
spe-
cial
fiber
of
the
p-adic
curve
under
consideration.
By
combining
this
notion
with
combinatorial
anabelian
results
and
techniques
de-
veloped
in
earlier
papers
by
the
authors,
together
with
the
theory
of
cyclotomic
synchronization
[also
developed
in
earlier
papers
by
the
authors],
we
obtain
a
generalization
of
André’s
characterization
of
the
local
Galois
groups
in
the
global
Galois
image
associated
to
a
hyperbolic
curve
to
the
case
of
arbitrary
hyperbolic
curves
[cf.
(a)].
Moreover,
by
applying
the
theory
of
local
contractibility
of
p-adic
analytic
spaces
developed
by
Berkovich,
we
show
that
the
techniques
developed
in
the
present
and
earlier
papers
by
the
authors
allow
one
to
relate
the
groups
of
M-admissible
outer
automorphisms
treated
in
the
present
paper
to
the
groups
of
outer
automorphisms
2010
Mathematics
Subject
Classification.
Primary
14H30;
Secondary
14H10.
Key
words
and
phrases.
anabelian
geometry,
combinatorial
anabelian
geometry,
tempered
fundamental
group,
tripod,
Grothendieck-Teichmüller
group,
semi-graph
of
anabelioids,
hyperbolic
curve,
configuration
space.
The
first
author
was
supported
by
Grant-in-Aid
for
Scientific
Research
(C),
No.
24540016,
Japan
Society
for
the
Promotion
of
Science.
1
2
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
of
tempered
fundamental
groups
of
higher-dimensional
configuration
spaces
[associated
to
the
given
p-adic
hyperbolic
curve].
These
con-
siderations
allow
one
to
“repair”
the
gap
in
André’s
proof
—
albeit
at
the
expense
of
working
with
M-admissible
outer
automorphisms
—
and
hence
to
realize
the
goal
of
obtaining
a
“local
analogue
of
the
Grothendieck-Teichmüller
group”
[cf.
(b)].
Contents
Introduction
2
0.
Notations
and
Conventions
11
1.
Almost
pro-Σ
combinatorial
anabelian
geometry
12
2.
Almost
pro-Σ
injectivity
26
3.
Applications
to
the
theory
of
tempered
fundamental
groups
54
References
103
Introduction
Let
Σ
⊆
Primes
be
a
subset
of
the
set
of
prime
numbers
Primes
which
is
either
equal
to
Primes
or
of
cardinality
one.
In
the
present
paper,
we
continue
our
study
of
the
pro-Σ
fundamental
groups
of
hyper-
bolic
curves
and
their
associated
configuration
spaces
over
algebraically
closed
fields
in
which
the
primes
of
Σ
are
invertible
[cf.
[MzTa],
[CmbCsp],
[NodNon],
[CbTpI],
[CbTpII]].
The
focus
of
the
present
paper
is
on
ap-
plications
of
the
theory
developed
in
previous
papers
to
the
theory
of
tempered
fundamental
groups,
in
the
style
of
[André].
Just
as
in
previous
papers,
the
main
technical
result
that
underlies
our
approach
is
a
certain
combinatorial
anabelian
result
[cf.
Theo-
rem
1.11;
Corollary
1.12],
which
may
be
summarized
as
a
generaliza-
tion
of
results
obtained
in
earlier
papers
[cf.,
e.g.,
[NodNon],
Theorem
A;
[CbTpII],
Theorem
1.9]
in
the
case
of
pro-Σ
fundamental
groups
to
the
case
of
almost
pro-Σ
fundamental
groups
[i.e.,
maximal
almost
pro-Σ
quotients
of
profinite
fundamental
groups
—
cf.
Definition
1.1].
The
technical
details
surrounding
this
generalization
occupy
the
bulk
of
§1.
In
§2,
we
observe
that
the
theory
of
§1
may
be
applied,
via
a
similar
argument
to
the
argument
applied
in
[NodNon]
to
derive
[NodNon],
Theorem
B,
from
[NodNon],
Theorem
A,
to
obtain
almost
pro-Σ
gen-
eralizations
[cf.
Theorem
2.9;
Corollary
2.10;
Remark
2.10.1]
of
the
injectivity
portion
of
the
theory
of
combinatorial
cuspidalization
[i.e.,
[NodNon],
Theorem
B].
In
the
final
portion
of
§2,
we
discuss
the
theory
of
almost
pro-l
commensurators
of
tripods
[i.e.,
copies
of
the
[geomet-
ric
fundamental
group
of
the]
projective
line
minus
three
points
—
cf.
Lemma
2.12,
Corollary
2.13],
in
the
context
of
the
theory
of
the
tripod
homomorphism
developed
in
[CbTpII],
§3.
Just
as
in
the
case
of
the
COMBINATORIAL
ANABELIAN
TOPICS
III
3
theory
of
§1,
the
theory
of
§2
is
conceptually
not
very
difficult,
but
technically
quite
involved.
Before
proceeding,
we
recall
that
a
substantial
portion
of
the
theory
of
[André]
revolves
around
the
study
of
outomorphism
[i.e.,
outer
auto-
morphism]
groups
of
the
tempered
geometric
fundamental
group
of
a
p-adic
hyperbolic
curve,
from
the
point
of
view
of
the
goal
of
es-
tablishing
a
p-adic
local
analogue
of
the
well-known
theory
of
the
Grothendieck-Teichmüller
group
[i.e.,
which
appears
in
the
context
of
hyperbolic
curves
over
num-
ber
fields].
From
the
point
of
view
of
the
theory
of
the
present
series
of
papers,
out-
omorphisms
of
such
tempered
fundamental
groups
may
be
thought
of
as
[i.e.,
are
equivalent
to
—
cf.
Remark
3.3.1;
Proposition
3.6,
(iii);
Re-
mark
3.13.1,
(i)]
outomorphisms
of
the
profinite
geometric
fundamen-
tal
group
that
are
“G-admissible”
[cf.
Definition
3.7,
(i)],
i.e.,
preserve
the
graph-theoretic
structure
on
the
profinite
geometric
fundamental
group.
In
a
word,
the
essential
thrust
of
the
applications
to
the
theory
of
tempered
fundamental
groups
given
in
the
present
paper
may
be
summarized
as
follows:
By
replacing,
in
effect,
the
G-admissible
outomorphism
groups
that
[modulo
the
“translation”
discussed
above]
appear
throughout
the
theory
of
[André]
by
“M-ad-
missible”
outomorphism
groups
—
i.e.,
groups
of
out-
omorphisms
of
the
profinite
geometric
fundamental
group
that
preserve
not
only
the
graph-theoretic
struc-
ture
on
the
profinite
geometric
fundamental
group,
but
also
the
[somewhat
finer]
metric
structure
on
the
var-
ious
dual
graphs
that
appear
[i.e.,
the
various
indices
at
the
nodes
of
the
special
fiber
of
the
p-adic
curve
under
consideration
—
cf.
Definition
3.7,
(ii)]
—
it
is
possible
to
overcome
various
significant
technical
difficulties
that
appear
in
the
theory
of
[André].
Here,
we
recall
that
the
two
main
technical
difficulties
that
appear
in
the
theory
of
[André]
may
be
described
as
follows:
•
The
characterization
of
the
local
Galois
groups
in
the
global
Galois
image
associated
to
a
hyperbolic
curve
that
is
given
in
[André],
Theorems
7.2.1,
7.2.3,
is
only
proven
for
a
quite
limited
class
of
hyperbolic
curves
[i.e.,
a
class
that
is
“far
from
generic”
—
cf.
[MzTa],
Corollary
5.7],
which
are
“closely
related
to
tripods”.
•
The
proof
given
in
[André]
of
a
certain
key
injectivity
result,
which
is
of
central
importance
in
establishing
the
theory
of
a
4
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
“local
analogue
of
the
Grothendieck-Teichmüller
group”,
con-
tains
a
fundamental
gap
[cf.
Remark
3.19.1].
In
the
present
paper,
our
approach
to
surmounting
the
first
technical
difficulty
consists
of
the
following
result
[cf.
Theorems
3.17,
(iv);
3.18,
(i)],
which
asserts,
roughly
speaking,
that
the
theory
of
the
tripod
homomorphism
developed
in
[CbTpII],
§3,
is
compatible
with
the
property
of
M-admissibility.
Theorem
A
(Metric-admissible
outomorphisms
and
the
tripod
homomorphism).
Let
n
≥
3
be
an
integer;
(g,
r)
a
pair
of
nonnega-
tive
integers
such
that
2g
−
2
+
r
>
0;
p
a
prime
number;
Σ
a
set
of
prime
numbers
such
that
Σ
=
{p},
and,
moreover,
is
either
equal
to
the
set
of
all
prime
numbers
or
of
cardinality
one;
R
a
mixed
character-
istic
complete
discrete
valuation
ring
of
residue
characteristic
p
whose
residue
field
is
separably
closed;
K
the
field
of
fractions
of
R;
K
an
algebraic
closure
of
K;
log
X
K
a
smooth
log
curve
of
type
(g,
r)
over
K.
Write
(X
K
)
log
n
for
the
n-th
log
configuration
space
[cf.
the
discussion
entitled
def
log
log
over
K;
(X
K
)
log
“Curves”
in
[CbTpII],
§0]
of
X
K
n
=
(X
K
)
n
×
K
K;
def
Σ
Π
n
=
π
1
((X
K
)
log
n
)
for
the
maximal
pro-Σ
quotient
of
the
log
fundamental
group
of
tpd
be
a
1-central
{1,
2,
3}-tripod
of
Π
n
[cf.
[CbTpII],
(X
K
)
log
n
.
Let
Π
Definitions
3.3,
(i);
3.7,
(ii)].
Then
the
restriction
of
the
tripod
ho-
momorphism
associated
to
Π
n
T
Π
tpd
:
Out
FC
(Π
n
)
−→
Out
C
(Π
tpd
)
[cf.
[CbTpII],
Definition
3.19]
to
the
subgroup
Out
FC
(Π
n
)
M
⊆
Out
FC
(Π
n
)
of
M-admissible
outomorphisms
[cf.
Definition
3.7,
(iii)]
factors
through
the
subgroup
Out(Π
tpd
)
M
⊆
Out
C
(Π
tpd
)
[cf.
Definition
3.7,
(i),
(ii);
Remark
3.13.1,
(i),
(ii)],
i.e.,
we
have
a
natural
commuta-
tive
diagram
of
profinite
groups
Out
FC
(Π
n
)
M
−−−→
Out(Π
tpd
)
M
⏐
⏐
⏐
⏐
Out
FC
(Π
n
)
−−−→
Out
C
(Π
tpd
)
.
T
Πtpd
Theorem
A
has
the
following
formal
consequence,
namely,
a
gener-
alization
of
the
characterization
of
the
local
Galois
groups
in
the
global
Galois
image
associated
to
a
hyperbolic
curve
that
is
given
in
[André],
Theorems
7.2.1,
7.2.3,
to
arbitrary
hyperbolic
curves,
albeit
at
COMBINATORIAL
ANABELIAN
TOPICS
III
5
the
expense
of,
in
effect,
replacing
“G-admissibility”
by
the
stronger
condition
of
“M-admissibility”
[cf.
Corollary
3.20;
Remark
3.20.1].
This
generalization
may
also
be
regarded
as
a
sort
of
strong
version
of
the
Galois
injectivity
result
given
in
[NodNon],
Theorem
C
[cf.
Re-
mark
3.20.2].
Theorem
B
(Characterization
of
the
local
Galois
groups
in
the
global
Galois
image
associated
to
a
hyperbolic
curve).
Let
F
be
a
number
field,
i.e.,
a
finite
extension
of
the
field
of
rational
numbers;
p
a
nonarchimedean
prime
of
F
;
F
p
an
algebraic
closure
of
the
p-adic
completion
F
p
of
F
;
F
⊆
F
p
the
algebraic
closure
of
F
in
∧
F
p
;
X
F
log
a
smooth
log
curve
over
F
.
Write
F
p
for
the
completion
def
def
def
of
F
p
;
G
p
=
Gal(F
p
/F
p
)
⊆
G
F
=
Gal(F
/F
);
X
F
log
=
X
F
log
×
F
F
;
π
1
(X
F
log
)
for
the
log
fundamental
group
of
X
F
log
[which,
in
the
following,
we
∧
identify
with
the
log
fundamental
groups
of
X
F
log
×
F
F
p
,
X
F
log
×
F
F
p
—
cf.
the
definition
of
F
!];
∧
π
1
temp
(X
F
log
×
F
F
p
)
∧
for
the
tempered
fundamental
group
of
X
F
log
×
F
F
p
[cf.
[André],
§4];
ρ
X
log
:
G
F
−→
Out(π
1
(X
F
log
))
F
for
the
natural
outer
Galois
action
associated
to
X
F
log
;
∧
ρ
temp
:
G
p
−→
Out(π
1
temp
(X
F
log
×
F
F
p
))
X
log
,p
F
for
the
natural
outer
Galois
action
associated
to
X
F
log
×
F
F
p
[cf.
[André],
Proposition
5.1.1];
∧
Out(π
1
(X
F
log
))
M
⊆
(
Out(π
1
temp
(X
F
log
×
F
F
p
))
⊆
)
Out(π
1
(X
F
log
))
for
the
subgroup
of
M-admissible
outomorphisms
of
π
1
(X
F
log
)
[cf.
Def-
inition
3.7,
(i),
(ii);
Proposition
3.6,
(i)].
Then
the
following
hold:
factors
through
the
subgroup
(i)
The
outer
Galois
action
ρ
temp
X
log
,p
F
∧
Out(π
1
(X
F
log
))
M
⊆
Out(π
1
temp
(X
F
log
×
F
F
p
)).
(ii)
We
have
a
natural
commutative
diagram
G
p
−−−→
Out(π
1
(X
F
log
))
M
⏐
⏐
⏐
⏐
ρ
X
log
F
G
F
−−−
→
Out(π
1
(X
F
log
))
6
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
—
where
the
vertical
arrows
are
the
natural
inclusions,
the
upper
horizontal
arrow
is
the
homomorphism
arising
from
the
factorization
of
(i),
and
all
arrows
are
injective.
(iii)
The
diagram
of
(ii)
is
cartesian,
i.e.,
if
we
regard
the
various
groups
involved
as
subgroups
of
Out(π
1
(X
F
log
)),
then
we
have
an
equality
G
p
=
G
F
∩
Out(π
1
(X
F
log
))
M
.
One
central
technical
aspect
of
the
theory
of
the
present
paper
lies
in
the
equivalence
[cf.
Theorem
3.9]
between
the
M-admissibility
of
outomorphisms
of
the
profinite
geometric
fundamental
group
of
the
given
p-adic
hyperbolic
curve
and
the
I-admissibility
[i.e.,
roughly
speaking,
compatibility
with
the
outer
action,
by
some
open
subgroup
of
the
inertia
group
of
the
absolute
Galois
group
of
the
base
field,
on
an
arbitrary
almost
pro-l
quotient
of
the
profinite
geometric
fun-
damental
group
—
cf.
Definition
3.8]
of
such
outomorphisms.
This
equivalence
is
obtained
by
applying
the
theory
of
cyclotomic
syn-
chronization
developed
in
[CbTpI],
§5.
Once
this
equivalence
is
es-
tablished,
the
almost
pro-l
injectivity
results
obtained
in
§2
then
al-
low
us
to
conclude
that
this
M-admissibility
of
outomorphisms
of
the
profinite
geometric
fundamental
group
of
the
given
p-adic
hyperbolic
curve
is,
in
fact,
equivalent
to
the
I-admissibility
of
any
[necessarily
unique!]
lifting
of
such
an
outomorphism
to
an
outomorphism
of
the
profinite
geometric
fundamental
group
of
a
higher-dimensional
con-
figuration
space
associated
to
the
given
p-adic
hyperbolic
curve
[cf.
Theorem
3.17,
(ii)].
Finally,
by
combining
this
“higher-dimensional
I-admissibility”
with
the
combinatorial
anabelian
theory
of
[CbTpII],
§1,
we
conclude
[cf.
Proposition
3.16,
(i);
Theorem
3.17,
(ii)]
that
a
certain
“higher-dimensional
G-admissibility”
also
holds,
i.e.,
that
the
lifted
outomorphism
of
the
profinite
geometric
fundamental
group
of
a
higher-dimensional
configuration
space
associated
to
the
given
p-
adic
hyperbolic
curve
preserves
the
graph-theoretic
structure
not
only
on
the
profinite
geometric
fundamental
group
of
the
original
hyperbolic
curve,
but
also
on
the
profinite
geometric
fundamental
groups
of
the
various
successive
fibers
of
the
higher-dimensional
configuration
space
under
consideration.
In
a
word,
it
is
precisely
by
applying
this
chain
of
equivalences
—
which
allows
us
to
control
the
graph-theoretic
struc-
ture
of
the
successive
fibers
of
the
higher-dimensional
configuration
space
under
consideration
—
that
allow
us
to
surmount
the
two
main
technical
difficulties
dis-
cussed
above
that
appear
in
the
theory
of
[André].
Put
another
way,
if,
instead
of
considering
M-admissible
outomor-
phisms
[i.e.,
of
the
profinite
geometric
fundamental
group
of
the
given
COMBINATORIAL
ANABELIAN
TOPICS
III
7
p-adic
hyperbolic
curve],
one
considers
arbitrary
G-admissible
outo-
morphisms
[of
the
profinite
geometric
fundamental
group
of
the
given
p-adic
hyperbolic
curve,
as
is
done,
in
effect,
in
[André]],
then
there
does
not
appear
to
exist,
at
least
at
the
time
of
writing,
any
effective
way
to
control
the
graph-theoretic
structure
on
the
successive
fibers
of
higher-dimensional
configuration
spaces.
In
this
context,
we
recall
that
in
the
theory
of
[CbTpII],
a
result
is
obtained
concerning
the
preservation
of
the
graph-theoretic
structure
on
the
successive
fibers
of
higher-dimensional
configuration
spaces
[cf.
[CbTpII],
Theorem
4.7],
in
the
context
of
pro-l
geometric
fundamental
groups.
The
significance,
however,
of
the
theory
of
the
present
paper
is
that
it
may
be
applied
to
almost
pro-l
geometric
fundamental
groups,
i.e.,
where
the
order
of
the
finite
quotient
implicit
in
the
term
“almost”
is
allowed
to
be
divisible
by
p.
Once
one
establishes
the
“higher-dimensional
G-admissibility”
dis-
cussed
above,
it
is
then
possible
to
apply
the
theory
of
local
contractibil-
ity
of
p-adic
analytic
spaces
developed
in
[Brk]
to
construct
from
the
given
outomorphism
of
a
profinite
geometric
fundamental
group
[of
a
higher-dimensional
configuration
space]
an
outomorphism
of
the
corre-
sponding
tempered
fundamental
group
[cf.
Proposition
3.16,
(ii)].
This
portion
of
the
theory
may
be
summarized
as
follows
[cf.
Theorem
3.19,
(ii)].
Theorem
C
(Metric-admissible
outomorphisms
and
tempered
fundamental
groups).
Let
n
be
a
positive
integer;
(g,
r)
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
p
a
prime
number;
Σ
a
nonempty
set
of
prime
numbers
such
that
Σ
=
{p},
and,
moreover,
if
n
≥
2,
then
Σ
is
either
equal
to
the
set
of
all
prime
numbers
or
of
cardinality
one;
R
a
mixed
characteristic
complete
discrete
valuation
ring
of
residue
characteristic
p
whose
residue
field
is
separably
closed;
K
the
field
of
fractions
of
R;
K
an
algebraic
closure
of
K;
log
X
K
a
smooth
log
curve
of
type
(g,
r)
over
K.
Write
(X
K
)
log
n
for
the
n-th
log
configuration
space
[cf.
the
discussion
entitled
def
log
log
“Curves”
in
[CbTpII],
§0]
of
X
K
over
K;
(X
K
)
log
n
=
(X
K
)
n
×
K
K;
def
Σ
Π
n
=
π
1
((X
K
)
log
n
)
for
the
maximal
pro-Σ
quotient
of
the
log
fundamental
group
of
∧
(X
K
)
log
n
;
K
for
the
p-adic
completion
of
K;
∧
π
1
temp
((X
K
)
log
n
×
K
K
)
for
the
tempered
fundamental
group
[cf.
[André],
§4,
as
well
as
the
discussion
of
Definition
3.1,
(ii),
of
the
present
paper]
of
(X
K
)
log
n
×
K
8
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∧
K
;
∧
def
Π
tp
=
lim
π
1
temp
((X
K
)
log
n
n
×
K
K
)/N
←−
N
∧
for
the
Σ-tempered
fundamental
group
of
(X
K
)
log
[cf.
n
×
K
K
[CmbGC],
Corollary
2.10,
(iii)],
i.e.,
the
inverse
limit
given
by
allow-
∧
ing
N
to
vary
over
the
open
normal
subgroups
of
π
1
temp
((X
K
)
log
n
×
K
K
)
such
that
the
quotient
by
N
corresponds
to
a
topological
covering
[cf.
[André],
§4.2,
as
well
as
the
discussion
of
Definition
3.1,
(ii),
of
the
present
paper]
of
some
finite
log
étale
Galois
covering
of
∧
(X
K
)
log
of
degree
a
product
of
primes
∈
Σ.
[Here,
we
recall
n
×
K
K
that,
when
n
=
1,
such
a
“topological
covering”
corresponds
to
a
“com-
binatorial
covering”,
i.e.,
a
covering
determined
by
a
covering
of
the
dual
semi-graph
of
the
special
fiber
of
the
stable
model
of
some
finite
∧
log
étale
covering
of
(X
K
)
log
n
×
K
K
.]
Write
M
tp
Out
FC
(Π
tp
n
)
⊆
Out(Π
n
)
for
the
inverse
image
of
Out
FC
(Π
n
)
M
⊆
Out(Π
n
)
[cf.
Definition
3.7,
(iii)]
via
the
natural
homomorphism
Out(Π
tp
n
)
→
Out(Π
n
)
[cf.
Propo-
sition
3.3,
(i)].
Then
the
resulting
natural
homomorphism
FC
M
M
Out
FC
(Π
tp
n
)
−→
Out
(Π
n
)
is
split
surjective,
i.e.,
there
exists
a
homomorphism
M
Φ
:
Out
FC
(Π
n
)
M
−→
Out
FC
(Π
tp
n
)
such
that
the
composite
Φ
FC
M
M
Out
FC
(Π
n
)
M
−→
Out
FC
(Π
tp
n
)
−→
Out
(Π
n
)
is
the
identity
automorphism
of
Out
FC
(Π
n
)
M
.
Up
till
now,
in
the
present
discussion,
the
p-adic
hyperbolic
curve
un-
der
consideration
was
arbitrary.
If,
however,
one
specializes
the
theory
discussed
above
to
the
case
of
tripods
[i.e.,
copies
of
the
projective
line
minus
three
points],
then
one
obtains
the
desired
p-adic
local
analogue
of
the
theory
of
the
Grothendieck-Teichmüller
group,
by
considering
the
“metrized
Grothendieck-Teichmüller
group
GT
M
”
as
follows
[cf.
Theorem
3.17,
(iv);
Theorem
3.18,
(ii);
Theorem
3.19,
(ii);
Remarks
3.19.2,
3.20.3].
Theorem
D
(Metric-admissible
outomorphisms
and
tripods).
In
the
notation
of
Theorem
C,
suppose
that
(g,
r)
=
(0,
3).
Write
Out
F
(Π
n
)
Δ+
⊆
Out
F
(Π
n
)
for
the
inverse
image
via
the
natural
homomorphism
Out
F
(Π
n
)
→
Out(Π
1
)
[cf.
[CbTpI],
Theorem
A,
(i)]
of
Out
C
(Π
1
)
Δ+
⊆
Out(Π
1
)
COMBINATORIAL
ANABELIAN
TOPICS
III
9
[cf.
[CbTpII],
Definition
3.4,
(i)];
def
Out
FC
(Π
n
)
Δ+
=
Out
F
(Π
n
)
Δ+
∩
Out
FC
(Π
n
)
[cf.
Remark
3.18.1];
def
Out
F
(Π
n
)
MΔ+
=
Out
F
(Π
n
)
Δ+
∩
Out
F
(Π
n
)
M
;
def
Out
FC
(Π
n
)
MΔ+
=
Out
FC
(Π
n
)
Δ+
∩
Out
F
(Π
n
)
M
.
Then
the
following
hold:
(i)
We
have
equalities
Out
F
(Π
n
)
Δ+
=
Out
FC
(Π
n
)
Δ+
,
Out
F
(Π
n
)
MΔ+
=
Out
FC
(Π
n
)
MΔ+
.
Moreover,
the
natural
homomorphisms
of
profinite
groups
Out
FC
(Π
n+1
)
Δ+
−−−→
Out
FC
(Π
n
)
Δ+
Out
F
(Π
n+1
)
Δ+
−−−→
Out
F
(Π
n
)
Δ+
Out
FC
(Π
n+1
)
MΔ+
−−−→
Out
FC
(Π
n
)
MΔ+
Out
F
(Π
n+1
)
MΔ+
−−−→
Out
F
(Π
n
)
MΔ+
are
bijective
for
n
≥
1.
In
the
following,
we
shall
identify
the
various
groups
that
occur
for
varying
n
by
means
of
these
natural
isomorphisms
and
write
def
GT
M
=
Out
F
(Π
n
)
MΔ+
=
Out
FC
(Π
n
)
MΔ+
def
⊆
GT
=
Out
F
(Π
n
)
Δ+
=
Out
FC
(Π
n
)
Δ+
[cf.
[CmbCsp],
Remark
1.11.1].
(ii)
Write
MΔ+
⊆
Out(Π
tp
Out
FC
(Π
tp
n
)
n
)
for
the
inverse
image
of
GT
M
⊆
Out(Π
n
)
[cf.
(i)]
via
the
natu-
ral
homomorphism
Out(Π
tp
n
)
→
Out(Π
n
)
[cf.
Proposition
3.3,
(i)].
Then
the
resulting
natural
homomorphism
MΔ+
−→
GT
M
Out
FC
(Π
tp
n
)
is
split
surjective,
i.e.,
there
exists
a
homomorphism
MΔ+
Φ
GT
:
GT
M
−→
Out
FC
(Π
tp
n
)
such
that
the
composite
Φ
GT
MΔ+
GT
M
−→
Out
FC
(Π
tp
−→
GT
M
n
)
10
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
is
the
identity
automorphism
of
GT
M
.
In
closing,
we
recall
that
“conventional
research”
concerning
the
Grothendieck-Teichmüller
group
GT
tends
to
focus
on
the
issue
of
whether
or
not
the
natural
inclusion
of
the
absolute
Galois
group
of
Q
G
Q
→
GT
is,
in
fact,
an
isomorphism
[cf.
the
discussion
of
[CbTpII],
Remark
3.19.1].
By
contrast,
one
important
theme
of
the
present
series
of
papers
lies
in
the
point
of
view
that,
instead
of
pursuing
the
issue
of
whether
or
not
GT
is
literally
isomorphic
to
G
Q
,
it
is
perhaps
more
natural
to
concentrate
on
the
issue
of
verifying
that
GT
exhibits
analogous
behavior/properties
to
G
Q
[or
Q].
From
this
point
of
view,
the
theory
of
tripod
synchronization
and
surjectivity
of
the
tripod
homomorphism
developed
in
[CbTpII]
[cf.
[CbTpII],
Theorem
C,
(iii),
(iv),
as
well
as
the
following
discus-
sion]
may
be
regarded
as
an
abstract
combinatorial
analogue
of
the
scheme-theoretic
fact
that
Spec
Q
lies
under
all
characteristic
zero
schemes/algebraic
stacks
in
a
unique
fashion
—
i.e.,
put
another
way,
that
all
morphisms
between
schemes
and
moduli
stacks
that
occur
in
the
theory
of
hyperbolic
curves
in
characteristic
zero
are
compatible
with
the
respective
structure
morphisms
to
Spec
Q.
In
a
similar
vein,
the
theory
of
the
subgroup
GT
M
⊆
GT
developed
in
the
present
paper
may
be
regarded
as
an
abstract
combinatorial
analogue
of
the
various
decomposition
subgroups
G
p
⊆
G
F
(⊆
G
Q
)
[cf.
Theorem
B]
as-
sociated
to
nonarchimedean
primes.
In
particular,
from
the
point
of
view
of
pursuing
“abstract
behavioral
similarities”
to
the
subgroups
G
p
⊆
G
F
(⊆
G
Q
),
it
is
natural
to
pose
the
question:
Is
the
subgroup
GT
M
⊆
GT
commensurably
terminal?
Unfortunately,
in
the
present
paper,
we
are
only
able
to
give
a
partial
answer
to
this
question.
That
is
to
say,
we
show
[cf.
Theorem
3.17,
(v),
and
its
proof;
Remark
3.20.1]
the
following
result.
[Here,
we
remark
that
although
this
result
is
not
stated
explicitly
in
Theorem
3.17,
(v),
it
follows
by
applying
to
GT
M
the
argument,
involving
l-graphically
full
actions,
that
was
applied,
in
the
proof
of
Theorem
3.17,
(v),
to
“Out
FC
(Π
n
)
M
”.]
Theorem
E
(Commensurator
of
the
metrized
Grothendieck-
-Teichmüller
group).
In
the
notation
of
Theorem
D
[cf.,
especially,
the
bijections
of
Theorem
D,
(i)],
the
commensurator
of
GT
M
in
Out
F
(Π
n
)
is
contained
in
the
subgroup
Out
G
(Π
n
)
⊆
Out
FC
(Π
n
)
COMBINATORIAL
ANABELIAN
TOPICS
III
11
of
outomorphisms
that
satisfy
the
condition
of
“higher-dimensional
G-admissibility”
discussed
above
[cf.
Definition
3.13,
(iv);
Remark
3.13.1,
(ii)].
In
particular,
the
commensurator
of
GT
M
in
GT
is
contained
in
G
def
G
FC
Out
(Π
n
)
⊆
Out
(Π
n
)
⊆
Out(Π
1
)
GT
=
GT
∩
n≥1
n≥1
[cf.
the
injections
Out
FC
(Π
n+1
)
→
Out
FC
(Π
n
)
of
[NodNon],
Theorem
B].
Acknowledgment
The
authors
would
like
to
thank
E.
Lepage
for
helpful
discussions
concerning
the
theory
of
Berkovich
spaces
and
Y.
Iijima
for
informing
us
of
[Prs].
0.
Notations
and
Conventions
Topological
groups:
Let
G
be
a
profinite
group
and
Σ
a
nonempty
set
of
prime
numbers.
Then
we
shall
write
G
Σ
for
the
maximal
pro-Σ
quotient
of
G.
Let
G
be
a
profinite
group
and
G
Q,
Q
quotients
of
G.
Then
we
shall
say
that
the
quotient
Q
dominates
the
quotient
Q
if
the
natural
surjection
G
Q
factors
through
the
natural
surjection
G
Q.
12
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
1.
Almost
pro-Σ
combinatorial
anabelian
geometry
In
the
present
§1,
we
discuss
almost
pro-Σ
analogues
of
results
on
combinatorial
anabelian
geometry
developed
in
earlier
papers
of
the
authors.
In
particular,
we
obtain
almost
pro-Σ
analogues
of
combina-
torial
versions
of
the
Grothendieck
Conjecture
for
outer
representations
of
NN-
and
IPSC-type
[cf.
Theorem
1.11;
Corollary
1.12
below].
These
almost
pro-Σ
analogues
of
combinatorial
versions
of
the
Grothendieck
Conjecture
will
be
applied
in
§2,
together
with
various
standard
tech-
niques
of
combinatorial
anabelian
geometry,
to
obtain
almost
pro-Σ
analogues
of
certain
standard
injectivity
results
that
will
be
applied
in
§3
to
derive
fundamental
results
concerning
the
outer
representations
of
Galois
groups
that
arise
from
hyperbolic
curves
and
their
associated
configuration
spaces
over
p-adic
fields.
In
the
present
§1,
let
Σ
⊆
Σ
†
be
nonempty
sets
of
prime
numbers
and
G
a
semi-graph
of
anabelioids
of
pro-Σ
†
PSC-type.
Write
G
for
the
underlying
semi-graph
of
G,
Π
G
for
the
[pro-Σ
†
]
fundamental
group
of
G,
and
G
→
G
for
the
universal
covering
of
G
corresponding
to
Π
G
.
Definition
1.1.
Let
G
be
a
profinite
group,
N
⊆
G
a
normal
open
subgroup
of
G,
and
G
Q
a
quotient
of
G.
Then
we
shall
say
that
Q
is
the
maximal
almost
pro-Σ
quotient
of
G
with
respect
to
N
if
the
kernel
of
the
surjection
G
Q
is
the
kernel
of
N
N
Σ
[cf.
the
discussion
entitled
“Topological
groups”
in
§0],
i.e.,
Q
=
G/Ker(N
N
Σ
).
Thus,
Q
fits
into
an
exact
sequence
of
profinite
groups
1
−→
N
Σ
−→
Q
−→
G/N
−→
1
.
[Note
that
since
N
is
normal
in
G,
and
the
kernel
Ker(N
N
Σ
)
of
the
natural
surjection
N
N
Σ
is
characteristic
in
N
,
it
holds
that
Ker(N
N
Σ
)
is
normal
in
G.]
We
shall
say
that
Q
is
a
maximal
almost
pro-Σ
quotient
of
G
if
Q
is
the
maximal
almost
pro-Σ
quotient
of
G
with
respect
to
some
normal
open
subgroup
of
G.
Lemma
1.2
(Properties
of
maximal
almost
pro-Σ
quotients).
Let
G
be
a
profinite
group.
Then
the
following
hold.
(i)
Let
N
⊆
G
be
a
normal
open
subgroup
of
G
and
G
J
a
quotient
of
G.
Write
N
J
⊆
J
for
the
image
of
N
in
J.
[Thus,
N
J
is
a
normal
open
subgroup
of
J.]
Then
the
quotient
of
J
determined
by
the
maximal
almost
pro-Σ
quotient
[cf.
Defini-
tion
1.1]
of
G
with
respect
to
N
,
i.e.,
the
quotient
of
J
by
the
image
of
Ker(N
N
Σ
)
in
J,
is
the
maximal
almost
pro-Σ
quotient
of
J
with
respect
to
N
J
.
(ii)
Let
N
⊆
G
be
a
normal
open
subgroup
of
G
and
H
⊆
G
a
closed
subgroup
of
G.
If
the
natural
homomorphism
(N
∩
H)
Σ
→
N
Σ
COMBINATORIAL
ANABELIAN
TOPICS
III
13
is
injective,
then
the
image
of
H
in
the
maximal
almost
pro-
Σ
quotient
of
G
with
respect
to
N
is
the
maximal
almost
pro-Σ
quotient
of
H
with
respect
to
N
∩
H.
(iii)
Let
H
⊆
G
be
a
normal
closed
subgroup
of
G
and
H
H
∗
a
maximal
almost
pro-Σ
quotient
of
H.
Suppose
that
H
is
topo-
logically
finitely
generated.
Then
there
exists
a
maximal
almost
pro-Σ
quotient
H
H
∗∗
of
H
which
dominates
H
H
∗
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
such
that
the
kernel
of
H
H
∗∗
is
normal
in
G.
Proof.
Assertions
(i),
(ii)
follow
immediately
from
the
various
defini-
tions
involved.
Next,
we
verify
assertion
(iii).
Let
N
⊆
H
be
a
normal
open
subgroup
of
H
with
respect
to
which
H
∗
is
the
maximal
almost
pro-Σ
quotient
of
H.
Now
since
H
is
topologically
finitely
generated,
and
N
⊆
H
is
open,
it
follows
that
there
exists
a
characteristic
open
subgroup
J
⊆
H
such
that
J
⊆
N
.
Observe
that
since
H
is
normal
in
G,
and
J
is
characteristic
in
H,
it
holds
that
J
is
normal
in
G.
Thus,
if
we
write
H
∗∗
for
the
maximal
almost
pro-Σ
quotient
of
H
with
respect
to
J,
then
H
∗∗
satisfies
the
conditions
of
assertion
(iii).
This
completes
the
proof
of
assertion
(iii).
Definition
1.3.
Let
I
be
a
profinite
group
and
ρ
:
I
→
Aut(G)
⊆
Out(Π
G
)
a
continuous
homomorphism.
Then
we
shall
say
that
ρ
is
of
PIPSC-type
[where
the
“PIPSC”
stands
for
“potentially
IPSC”]
if
the
following
conditions
are
satisfied:
†
(i)
I
is
isomorphic
to
Z
Σ
as
an
abstract
profinite
group.
(ii)
there
exists
an
open
subgroup
J
⊆
I
such
that
the
restriction
of
ρ
to
J
is
of
IPSC-type
[cf.
[NodNon],
Definition
2.4,
(i)].
Lemma
1.4
(Profinite
Dehn
multi-twists
and
finite
étale
cov-
erings).
Let
α
∈
Out(Π
G
),
α
∈
Aut(Π
G
)
a
lifting
of
α,
and
H
→
G
a
connected
finite
étale
Galois
subcovering
of
G
→
G
such
that
α
pre-
serves
the
corresponding
open
subgroup
Π
H
⊆
Π
G
,
hence
induces
an
element
α
H
∈
Out(Π
H
).
Suppose
that
α
H
∈
Dehn(H)
[cf.
[CbTpI],
Definition
4.4].
Then
α
∈
Dehn(G).
Proof.
It
follows
immediately
from
[CmbGC],
Propositions
1.2,
(ii);
1.5,
(ii),
that
α
∈
Aut(G).
The
fact
that
α
∈
Dehn(G)
now
follows
from
[CmbGC],
Propositions
1.2,
(i);
1.5,
(i),
together
with
the
commensu-
rable
terminality
of
VCN-subgroups
of
Π
G
[cf.
[CmbGC],
Proposition
1.2,
(ii)]
and
the
slimness
of
verticial
subgroups
of
Π
G
[cf.
[CmbGC],
14
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Remark
1.1.3].
[Here,
we
recall
that
an
automorphism
of
a
slim
profi-
nite
group
is
equal
to
the
identity
if
and
only
if
it
preserves
and
induces
the
identity
on
an
open
subgroup.]
Lemma
1.5
(Outer
representations
of
VA-,
NN-,
PIPSC-type
and
finite
étale
coverings).
In
the
notation
of
Definition
1.3,
sup-
†
pose
that
I
is
isomorphic
to
Z
Σ
as
an
abstract
profinite
group;
let
ρ
J
:
J
→
Aut(Π
G
)
be
a
lifting
of
the
restriction
of
ρ
to
an
open
subgroup
J
⊆
I
and
H
→
G
a
connected
finite
étale
Galois
subcovering
of
G
→
G
such
that
the
action
of
J
on
Π
G
,
via
ρ
J
,
preserves
the
corresponding
open
subgroup
Π
H
⊆
Π
G
,
hence
induces
a
continuous
homomorphism
J
→
Aut(Π
H
).
Then
ρ
is
of
VA-type
[cf.
[NodNon],
Definition
2.4,
(ii),
as
well
as
Remark
1.5.1
below]
(respectively,
NN-type
[cf.
[NodNon],
Definition
2.4,
(iii)];
PIPSC-type
[cf.
Definition
1.3])
if
and
only
if
the
composite
J
→
Aut(Π
H
)
Out(Π
H
)
is
of
VA-type
(respectively,
NN-type;
PIPSC-type).
Proof.
Necessity
in
the
case
of
outer
representations
of
VA-type
(re-
spectively,
NN-type;
PIPSC-type)
follows
immediately
from
[NodNon],
Lemma
2.6,
(i)
(respectively,
[NodNon],
Lemma
2.6,
(i);
the
various
definitions
involved,
together
with
the
well-known
properness
of
the
moduli
stack
of
pointed
stable
curves
of
a
given
type).
To
verify
suf-
ficiency,
let
us
first
observe
that
it
follows
immediately
from
the
vari-
ous
definitions
involved
that
we
may
assume
without
loss
of
generality
that
J
=
I,
and
that
the
outer
representation
J
=
I
→
Out(Π
H
)
is
of
SVA-type
(respectively,
SNN-type;
IPSC-type)
[cf.
[NodNon],
Def-
inition
2.4].
Then
sufficiency
in
the
case
of
outer
representations
of
VA-type
(respectively,
NN-type;
PIPSC-type)
follows
immediately,
in
light
of
the
criterion
of
[CbTpI],
Corollary
5.9,
(i)
(respectively,
(ii);
(iii)),
from
Lemma
1.4,
together
with
the
compatibility
property
of
[CbTpI],
Corollary
5.9,
(v)
[applied,
via
[CbTpI],
Theorem
4.8,
(ii),
(iv),
to
each
of
the
Dehn
coordinates
of
the
profinite
Dehn
multi-twists
under
consideration
—
cf.
the
proof
of
[CbTpII],
Lemma
3.26,
(ii)].
This
completes
the
proof
of
Lemma
1.5.
Remark
1.5.1.
Here,
we
take
the
opportunity
to
correct
an
unfor-
tunate
misprint
in
[NodNon].
The
phrase
“of
VA-type”
that
appears
near
the
beginning
of
[NodNon],
Definition
2.4,
(ii),
should
read
“is
of
VA-type”.
Definition
1.6.
Let
H
be
a
semi-graph
of
anabelioids
of
pro-Σ
†
PSC-
type.
Write
H
for
the
underlying
semi-graph
of
H,
Π
H
for
the
[pro-Σ
†
]
→
H
for
the
universal
covering
of
H
fundamental
group
of
H,
and
H
COMBINATORIAL
ANABELIAN
TOPICS
III
15
corresponding
to
Π
H
.
Let
Π
∗G
(respectively,
Π
∗H
)
be
a
maximal
almost
pro-Σ
quotient
of
Π
G
(respectively,
Π
H
)
[cf.
Definition
1.1].
(i)
For
each
v
∈
Vert(G)
(respectively,
e
∈
Edge(G);
e
∈
Node(G);
e
∈
Cusp(G);
z
∈
VCN(G)),
we
shall
refer
to
the
image
of
a
ver-
ticial
(respectively,
an
edge-like;
a
nodal;
a
cuspidal;
a
VCN-
[cf.
[CbTpI],
Definition
2.1,
(i)])
subgroup
of
Π
G
associated
to
v
(respectively,
e;
e;
e;
z)
in
the
quotient
Π
∗G
as
a
verticial
(respectively,
an
edge-like;
a
nodal;
a
cuspidal;
a
VCN-)
sub-
group
of
Π
∗G
associated
to
v
(respectively,
e;
e;
e;
z).
For
each
(respectively,
e
∈
Edge(
G);
e
∈
Node(
G);
element
v
∈
Vert(
G)
z
∈
VCN(
G)),
we
shall
refer
to
the
image
of
the
e
∈
Cusp(
G);
verticial
(respectively,
edge-like;
nodal;
cuspidal;
VCN-)
sub-
group
of
Π
G
associated
to
v
(respectively,
e
;
e
;
e
;
z
)
in
the
quotient
Π
∗G
as
the
verticial
(respectively,
edge-like;
nodal;
cus-
pidal;
VCN-)
subgroup
of
Π
∗G
associated
to
v
(respectively,
e
;
e
;
e
;
z
).
∼
(ii)
We
shall
say
that
an
isomorphism
Π
∗G
→
Π
∗H
is
group-theoreti-
cally
verticial
(respectively,
group-theoretically
nodal;
group-
theoretically
cuspidal)
if
the
isomorphism
induces
a
bijection
between
the
set
of
the
verticial
(respectively,
nodal;
cuspidal)
subgroups
[cf.
(i)]
of
Π
∗G
and
the
set
of
the
verticial
(respec-
tively,
nodal;
cuspidal)
subgroups
of
Π
∗H
.
We
shall
say
that
∼
an
outer
isomorphism
Π
∗G
→
Π
∗H
is
group-theoretically
verti-
cial
(respectively,
group-theoretically
nodal;
group-theoretically
cuspidal)
if
the
outer
isomorphism
arises
from
an
isomorphism
∼
∗
Π
∗G
→
Π
H
which
is
group-theoretically
verticial
(respectively,
group-theoretically
nodal;
group-theoretically
cuspidal).
∼
(iii)
We
shall
say
that
an
isomorphism
Π
∗G
→
Π
∗H
is
group-theoreti-
cally
graphic
if
the
isomorphism
is
group-theoretically
verti-
cial,
group-theoretically
nodal,
and
group-theoretically
cuspi-
∼
dal
[cf.
(ii)].
We
shall
say
that
an
outer
isomorphism
Π
∗G
→
Π
∗H
is
group-theoretically
graphic
if
the
outer
isomorphism
arises
∼
from
an
isomorphism
Π
∗G
→
Π
∗H
which
is
group-theoretically
graphic.
We
shall
write
Aut
grph
(Π
∗G
)
⊆
Aut(Π
∗G
)
for
the
subgroup
of
group-theoretically
graphic
automorphisms
of
Π
∗G
and
Out
grph
(Π
∗G
)
=
Aut
grph
(Π
∗G
)/Inn(Π
∗G
)
⊆
Out(Π
∗G
)
def
for
the
subgroup
of
group-theoretically
graphic
outomorphisms
of
Π
∗G
.
16
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(iv)
Let
I
be
a
profinite
group.
Then
we
shall
say
that
a
continuous
homomorphism
ρ
:
I
→
Aut
grph
(Π
∗G
)
⊆
Aut(Π
∗G
)
[cf.
(iii)]
is
of
VA-type
(respectively,
NN-type;
PIPSC-type)
if
the
following
condition
is
satisfied:
Let
N
⊆
Π
G
be
a
normal
open
subgroup
of
Π
G
with
respect
to
which
Π
∗G
is
the
maximal
almost
pro-
Σ
quotient
of
Π
G
.
[Thus,
N
Σ
⊆
Π
∗G
.]
Then
there
exists
a
characteristic
open
subgroup
M
⊆
Π
∗G
of
Π
∗G
such
that
the
following
conditions
are
satisfied:
(1)
M
⊆
N
Σ
.
[Thus,
M
may
be
regarded
as
the
[pro-Σ]
funda-
Σ
—
cf.
[SemiAn],
mental
group
of
the
pro-Σ
completion
G
M
Definition
2.9,
(ii)
—
of
the
connected
finite
étale
Galois
subcovering
G
M
→
G
of
G
→
G
corresponding
to
M
⊆
Π
∗G
,
Σ
.]
i.e.,
M
=
Π
G
M
Σ
),
(2)
The
composite
I
→
Aut(M
)
Out(M
)
=
Out(Π
G
M
where
the
first
arrow
is
the
homomorphism
induced
by
ρ,
is
of
VA-type
(respectively,
NN-type;
PIPSC-type)
in
the
sense
of
[NodNon],
Definition
2.4,
(ii)
[cf.
also
Re-
mark
1.5.1
of
the
present
paper]
(respectively,
[NodNon],
Definition
2.4,
(iii);
Definition
1.3
of
the
present
paper)
[i.e.,
as
an
outer
representation
of
pro-Σ
PSC-type
—
cf.
[NodNon],
Definition
2.1,
(i)].
[Here,
we
observe
that
it
follows
immediately
from
Lemma
1.5
that
condition
(2)
is
independent
of
the
choice
of
M
—
cf.
Lemma
1.9
below.]
We
shall
say
that
a
continuous
homomor-
phism
ρ
:
I
→
Out
grph
(Π
∗G
)
⊆
Out(Π
∗G
)
[cf.
(iii)]
is
of
VA-type
(respectively,
NN-type;
PIPSC-type)
if
ρ
arises
from
a
homo-
morphism
I
→
Aut
grph
(Π
∗G
)
⊆
Aut(Π
∗G
)
which
is
of
VA-type
(respectively,
NN-type;
PIPSC-type).
[Here,
we
observe
that
it
follows
immediately
from
Lemma
1.5,
together
with
the
slim-
ness
of
Π
∗G
[cf.
Proposition
1.7,
(i),
below],
that
this
condition
on
ρ
:
I
→
Out
grph
(Π
∗G
)
is
independent
of
the
choice
of
the
ho-
momorphism
I
→
Aut
grph
(Π
∗G
).]
(v)
Let
α
∈
Out(Π
∗G
).
Then
we
shall
say
that
α
is
a
profinite
Dehn
there
exists
a
lifting
multi-twist
of
Π
∗G
if,
for
each
v
∈
Vert(
G),
∗
α[
v
]
∈
Aut(Π
G
)
of
α
which
preserves
the
verticial
subgroup
∗
[cf.
(i)]
and
induces
the
Π
v
⊆
Π
∗G
associated
to
v
∈
Vert(
G)
∗
identity
automorphism
of
Π
v
.
We
shall
write
Dehn(Π
∗G
)
⊆
Out(Π
∗G
)
for
the
subgroup
of
profinite
Dehn
multi-twists
of
Π
∗G
.
COMBINATORIAL
ANABELIAN
TOPICS
III
17
Remark
1.6.1.
In
the
notation
of
Definition
1.6,
if
Π
∗G
,
Π
∗H
are
the
respective
maximal
almost
pro-Σ
quotients
of
Π
G
,
Π
H
with
respect
to
Π
G
,
Π
H
,
then
it
follows
immediately
from
the
various
definitions
involved
that
Π
∗G
,
Π
∗H
are
the
respective
maximal
pro-Σ
quotients
of
Π
G
,
Π
H
.
In
particular,
it
follows
immediately
that
one
may
regard
Π
∗G
,
Π
∗H
as
the
[pro-Σ]
fundamental
groups
of
the
semi-graphs
of
anabelioids
of
pro-Σ
PSC-type
G
Σ
,
H
Σ
obtained
by
forming
the
pro-Σ
completions
[cf.
[SemiAn]
Definition
2.9,
(ii)]
of
G,
H,
respectively,
i.e.,
Π
∗G
=
Π
G
Σ
,
Π
∗H
=
Π
H
Σ
.
Moreover,
one
verifies
immediately
that,
relative
to
these
identifications,
the
notions
defined
in
Definition
1.6,
(i),
(ii),
(iii),
(iv),
are
compatible
with
their
counterparts
defined
[for
the
most
part]
in
earlier
papers
of
the
authors:
•
VCN-subgroups
[cf.
[CbTpI],
Definition
2.1,
(i)];
•
group-theoretically
verticial/nodal/cuspidal/graphic
(outer)
iso-
morphisms
[cf.
[CmbGC],
Definition
1.4,
(i),
(iv);
[NodNon],
Definition
1.12];
•
outer
representations
of
VA-/NN-/PIPSC-type
[cf.
[NodNon],
Definition
2.4,
(ii),
(iii);
Remark
1.5.1
of
the
present
paper;
Definition
1.3
of
the
present
paper;
Lemma
1.5
of
the
present
paper];
•
profinite
Dehn
multi-twists
[cf.
[CbTpI],
Definition
4.4],
i.e.,
so
Dehn(G
Σ
)
=
Dehn(Π
∗G
)
⊆
Out
grph
(Π
∗G
).
Remark
1.6.2.
In
the
situation
of
Definition
1.6,
(iv),
it
follows
imme-
diately
from
Lemma
1.5,
together
with
[NodNon],
Remark
2.4.2,
that
we
have
implications
PIPSC-type
=⇒
NN-type
=⇒
VA-type
.
Proposition
1.7
(Properties
of
VCN-subgroups).
Let
Π
∗G
be
a
maximal
almost
pro-Σ
quotient
of
Π
G
[cf.
Definition
1.1].
For
v
,
e
∈
Edge(
G),
write
G
∗
→
G
for
the
connected
profinite
w
∈
Vert(
G);
étale
subcovering
of
G
→
G
corresponding
to
Π
∗G
;
Vert(G
∗
)
=
lim
Vert(G
),
Edge(G
∗
)
=
lim
Edge(G
)
←−
←−
—
where
the
projective
limits
range
over
all
connected
finite
étale
sub-
coverings
G
→
G
of
G
∗
→
G;
def
def
v
(G
∗
)
∈
Vert(G
∗
),
e
(G
∗
)
∈
Edge(G
∗
)
e
∈
Edge(
G)
via
the
natural
maps
for
the
images
of
v
∈
Vert(
G),
∗
∗
Vert(
G)
Vert(G
),
Edge(
G)
Edge(G
),
respectively;
∗
E
G
∗
:
Vert(G
∗
)
−→
2
Edge(G
)
18
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[cf.
the
discussion
entitled
“Sets”
in
[CbTpI],
§0,
concerning
the
no-
∗
tation
2
Edge(G
)
]
for
the
map
induced
by
the
various
E’s
involved
[cf.
[NodNon],
Definition
1.1,
(iv)];
δ(
v
(G
∗
),
w(G
∗
))
=
sup
{δ(
v
(G
),
w(G
))}
∈
N
∪
{∞}
def
G
[cf.
[NodNon],
Definition
1.1,
(vii)]
—
where
G
ranges
over
the
con-
nected
finite
étale
subcoverings
G
→
G
of
G
∗
→
G.
Then
the
following
hold:
(i)
Π
∗G
is
topologically
finitely
generated,
slim
[cf.
the
discus-
sion
entitled
“Topological
groups”
in
[CbTpI],
§0],
and
almost
torsion-free
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0].
In
particular,
every
VCN-subgroup
of
Π
∗G
[cf.
Definition
1.6,
(i)]
is
almost
torsion-free.
(ii)
Let
z
∈
VCN(G)
and
Π
z
⊆
Π
G
a
VCN-subgroup
of
Π
G
associ-
ated
to
z
∈
VCN(G).
Write
Π
∗
z
⊆
Π
G
∗
for
the
VCN-subgroup
of
Π
∗G
obtained
by
forming
the
image
of
Π
z
⊆
Π
G
in
Π
∗G
.
Then
Π
∗
z
is
a
maximal
almost
pro-Σ
quotient
of
Π
z
.
In
partic-
ular,
every
verticial
subgroup
of
Π
∗G
is
topologically
finitely
generated
and
slim.
Write
Π
∗
⊆
Π
∗
for
the
verticial
(iii)
For
i
=
1,
2,
let
v
i
∈
Vert(
G).
G
v
i
∗
subgroup
of
Π
G
associated
to
v
i
.
Consider
the
following
three
[mutually
exclusive]
conditions:
(1)
δ(
v
1
(G
∗
),
v
2
(G
∗
))
=
0.
(2)
δ(
v
1
(G
∗
),
v
2
(G
∗
))
=
1.
(3)
δ(
v
1
(G
∗
),
v
2
(G
∗
))
≥
2.
Then
we
have
equivalences
(1)
⇐⇒
(1
)
;
(2)
⇐⇒
(2
)
;
(3)
⇐⇒
(3
)
with
the
following
three
conditions:
(1
)
Π
∗
v
1
=
Π
∗
v
2
.
(2
)
Π
∗
v
1
∩
Π
∗
v
2
is
infinite,
but
Π
∗
v
1
=
Π
∗
v
2
.
(3
)
Π
∗
v
1
∩
Π
∗
v
2
is
finite.
(iv)
In
the
situation
of
(iii),
if
condition
(2),
hence
also
condition
(2
),
holds,
then
it
holds
that
(E
G
∗
(
v
1
(G
∗
))
∩
E
G
∗
(
v
2
(G
∗
)))
=
1,
∗
∗
∗
and,
moreover,
Π
v
1
∩
Π
v
2
=
Π
e
,
for
any
element
e
∈
Edge(
G)
∗
∗
∗
such
that
e
(G
)
∈
E
G
∗
(
v
1
(G
))
∩
E
G
∗
(
v
2
(G
)).
Write
Π
∗
⊆
Π
∗
for
the
edge-like
(v)
For
i
=
1,
2,
let
e
i
∈
Edge(
G).
G
e
i
subgroup
of
Π
∗G
associated
to
e
i
.
Then
Π
∗
e
1
∩
Π
∗
e
2
is
infinite
if
COMBINATORIAL
ANABELIAN
TOPICS
III
19
and
only
if
e
1
(G
∗
)
=
e
2
(G
∗
).
In
particular,
Π
∗
e
1
∩Π
∗
e
2
is
infinite
if
and
only
if
Π
∗
e
1
=
Π
∗
e
2
.
e
∈
Edge(
G).
Write
Π
∗
,
Π
∗
⊆
Π
∗
for
the
(vi)
Let
v
∈
Vert(
G),
G
v
e
VCN-subgroups
of
Π
∗G
associated
to
v
,
e
,
respectively.
Then
Π
∗
e
∩
Π
∗
v
is
infinite
if
and
only
if
e
(G
∗
)
∈
E
G
∗
(
v
(G
∗
)).
In
par-
ticular,
Π
∗
e
∩
Π
∗
v
is
infinite
if
and
only
if
Π
∗
e
⊆
Π
∗
v
.
(vii)
Every
VCN-subgroup
of
Π
∗G
is
commensurably
terminal
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0]
in
Π
∗G
.
(viii)
Let
z
∈
VCN(G),
Π
z
⊆
Π
G
a
VCN-subgroup
of
Π
G
associated
to
z
∈
VCN(G),
and
Π
z
Π
‡
z
an
almost
pro-Σ
quotient
of
Π
‡
z
.
Then
there
exists
a
maximal
almost
pro-Σ
quotient
Π
∗∗
G
of
Π
G
such
that
the
quotient
of
Π
z
determined
by
the
quo-
‡
tient
Π
G
Π
∗∗
G
dominates
the
quotient
Π
z
Π
z
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
Proof.
Let
N
⊆
Π
G
be
a
normal
open
subgroup
of
Π
G
with
respect
to
which
Π
∗G
is
the
maximal
almost
pro-Σ
quotient
of
Π
G
.
[Thus,
N
Σ
⊆
Π
∗G
.]
Write
G
N
→
G
for
the
connected
finite
étale
Galois
sub-
covering
of
G
→
G
corresponding
to
N
⊆
Π
G
.
Thus,
N
Σ
⊆
Π
∗G
may
be
regarded
as
the
[pro-Σ]
fundamental
group
of
the
pro-Σ
completion
Σ
G
N
[cf.
[SemiAn],
Definition
2.9,
(ii)]
of
G
N
,
i.e.,
N
Σ
=
Π
G
N
Σ
.
First,
we
verify
assertion
(i).
Since
Π
G
is
topologically
finitely
gen-
erated,
it
is
immediate
that
Π
∗G
is
topologically
finitely
generated.
Now
let
us
recall
[cf.
[MzTa],
Remark
1.2.2;
[MzTa],
Proposition
1.4]
that
N
Σ
=
Π
G
N
Σ
is
torsion-free
and
slim.
Thus,
the
fact
that
Π
∗G
is
almost
torsion-free
is
immediate;
the
slimness
of
Π
∗G
follows
immediately,
by
considering
the
natural
outer
action
Π
G
/N
→
Out(N
Σ
),
from
the
well-
known
fact
that
any
nontrivial
automorphism
of
a
stable
log
curve
over
an
algebraically
closed
field
of
characteristic
∈
Σ
induces
a
nontrivial
outomorphism
of
the
maximal
pro-Σ
quotient
of
the
geometric
log
fun-
damental
group
of
the
stable
curve
[cf.
[CmbGC],
Proposition
1.2,
(i),
(ii);
[MzTa],
Proposition
1.4,
applied
to
the
verticial
subgroups
of
the
geometric
log
fundamental
group
under
consideration].
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Let
us
recall
that
since
Π
z
∩
N
⊆
N
=
Π
G
N
is
a
VCN-subgroup
of
Π
G
N
,
the
natural
homomorphism
(Π
z
∩
N
)
Σ
→
Π
Σ
G
N
is
injective
[cf.,
e.g.,
the
proof
of
[SemiAn],
Proposition
2.5,
(i);
[SemiAn],
Example
2.10].
Thus,
it
follows
immediately
from
Lemma
1.2,
(ii),
that
Π
∗
z
is
a
maximal
almost
pro-Σ
quotient
of
Π
z
.
In
particular,
if
z
∈
Vert(G),
then
it
follows
immediately
from
assertion
(i)
that
Π
∗
z
is
topologically
finitely
generated
and
slim.
This
completes
the
proof
of
assertion
(ii).
20
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Next,
we
verify
assertions
(iii),
(v),
and
(vi).
Since
N
Σ
=
Π
G
N
Σ
,
one
verifies
easily
—
by
considering
the
intersections
of
N
Σ
=
Π
G
N
Σ
with
the
various
VCN-subgroups
of
Π
∗G
under
consideration
and
ap-
plying
[NodNon],
Lemma
1.9,
(ii)
(respectively,
[NodNon],
Lemma
1.5;
[NodNon],
Lemma
1.7),
together
with
the
well-known
fact
that
every
VCN-subgroup
of
Π
G
N
Σ
is
nontrivial
and
torsion-free
[hence
also
infinite]
—
that
assertion
(iii)
(respectively,
(v);
(vi))
holds.
This
completes
the
proof
of
assertions
(iii),
(v),
and
(vi).
Assertion
(vii)
follows
formally
from
assertions
(iii),
(v).
Indeed,
let
Π
∗
z
⊆
Π
∗G
be
the
VCN-subgroup
and
γ
∈
C
Π
∗
(Π
∗
).
Then
of
Π
∗G
associated
to
an
element
z
∈
VCN(
G)
z
G
it
follows
immediately
from
assertions
(iii),
(v)
that
z
=
z
γ
;
we
thus
conclude
that
γ
∈
Π
∗
z
.
This
completes
the
proof
of
assertion
(vii).
Next,
we
verify
assertion
(iv).
By
applying
[NodNon],
Lemma
1.8,
Σ
to
G
N
,
one
verifies
immediately
that
(E
G
∗
(
v
1
(G
∗
))
∩
E
G
∗
(
v
2
(G
∗
)))
=
1.
Thus,
it
follows
immediately
from
assertion
(vi);
[NodNon],
Lemma
1.5;
[NodNon],
Lemma
1.9,
(ii),
that
Π
∗
e
∩N
Σ
=
Π
∗
v
1
∩Π
∗
v
2
∩N
Σ
.
Since
N
Σ
is
open
in
Π
∗G
,
we
conclude
from
assertion
(vi)
that
Π
∗
e
is
an
open
subgroup
of
Π
∗
v
1
∩
Π
∗
v
2
,
hence
that
Π
∗
v
1
∩
Π
∗
v
2
⊆
C
Π
∗G
(Π
∗
e
)
=
Π
∗
e
[cf.
assertion
(vii)].
This
completes
the
proof
of
assertion
(iv).
Finally,
we
verify
assertion
(viii).
It
follows
from
the
definition
of
an
almost
pro-Σ
quotient
that
the
natural
surjection
Π
z
Π
‡
z
factors
through
a
maximal
almost
pro-Σ
quotient
of
Π
z
.
Thus,
by
replacing
Π
‡
z
by
a
suitable
maximal
almost
pro-Σ
quotient
of
Π
z
,
we
may
assume
without
loss
of
generality
that
Π
‡
z
is
a
maximal
almost
pro-Σ
quotient
of
Π
z
.
Let
N
z
⊆
Π
z
be
a
normal
open
subgroup
of
Π
z
with
respect
to
which
Π
‡
z
is
the
maximal
almost
pro-Σ
quotient
of
Π
z
and
N
G
⊆
Π
G
a
normal
open
subgroup
of
Π
G
such
that
N
G
∩
Π
z
⊆
N
z
.
Here,
we
recall
that
the
existence
of
such
a
subgroup
N
G
follows
immediately
from
the
fact
that
the
natural
profinite
topology
on
Π
z
coincides
with
the
topology
on
Π
z
induced
by
the
topology
of
Π
G
.
Then
one
verifies
easily
that
the
maximal
almost
pro-Σ
quotient
of
Π
G
with
respect
to
N
G
⊆
Π
G
is
a
maximal
almost
pro-Σ
quotient
of
Π
G
as
in
the
statement
of
assertion
(viii).
This
completes
the
proof
of
assertion
(viii).
Definition
1.8.
Let
Π
∗G
be
a
maximal
almost
pro-Σ
quotient
of
Π
G
[cf.
Definition
1.1].
Then
we
shall
write
Aut
|grph|
(Π
∗G
)
⊆
Aut
grph
(Π
∗G
)
for
the
subgroup
of
group-theoretically
graphic
[cf.
Definition
1.6,
(iii)]
automorphisms
α
of
Π
∗G
such
that
the
natural
action
of
α
on
the
un-
derlying
semi-graph
G
[determined
by
the
group-theoretic
graphicity
COMBINATORIAL
ANABELIAN
TOPICS
III
21
of
α,
together
with
Proposition
1.7,
(iii),
(v),
(vi)]
is
the
identity
auto-
morphism.
Also,
we
shall
write
Out
|grph|
(Π
∗G
)
=
Aut
|grph|
(Π
∗G
)/Inn(Π
∗G
)
⊆
Out(Π
∗G
)
.
def
for
the
image
of
Aut
|grph|
(Π
∗G
)
in
Out(Π
∗G
).
Remark
1.8.1.
In
the
notation
of
Definition
1.8,
one
verifies
easily
that
Dehn(Π
∗G
)
⊆
Out
|grph|
(Π
∗G
)
[cf.
Definitions
1.6,
(v);
1.8;
[CmbGC],
Proposition
1.2,
(i)].
Remark
1.8.2.
In
the
spirit
of
Remark
1.6.1,
one
verifies
immediately
that
the
notation
of
Definition
1.8
is
consistent
with
the
the
notation
of
[CbTpI],
Definition
2.6,
(i)
[cf.
also
[CbTpII],
Remark
4.1.2].
Lemma
1.9
(Alternative
characterization
of
outer
representa-
tions
of
VA-,
NN-,
PIPSC-type).
Let
Π
∗G
be
a
maximal
almost
pro-Σ
quotient
of
Π
G
[cf.
Definition
1.1],
I
a
profinite
group,
and
ρ
:
I
→
Aut
grph
(Π
∗G
)
a
continuous
homomorphism.
Then
the
following
conditions
are
equivalent:
(i)
ρ
is
of
VA-type
(respectively,
NN-type;
PIPSC-type)
[cf.
Definition
1.6,
(iv)].
(ii)
Let
N
⊆
Π
G
be
a
normal
open
subgroup
of
Π
G
with
respect
to
which
Π
∗G
is
the
maximal
almost
pro-Σ
quotient
of
Π
G
.
[Thus,
N
Σ
⊆
Π
∗G
.]
Let
M
⊆
Π
∗G
be
a
characteristic
open
subgroup
of
Π
∗G
such
that
M
⊆
N
Σ
.
[Thus,
M
may
be
regarded
as
Σ
the
[pro-Σ]
fundamental
group
of
the
pro-Σ
completion
G
M
—
cf.
[SemiAn],
Definition
2.9,
(ii)
—
of
the
connected
finite
étale
Galois
subcovering
G
M
→
G
of
G
→
G
corresponding
to
Σ
.]
M
⊆
Π
∗G
,
i.e.,
M
=
Π
G
M
Then
it
holds
that
the
compos-
Σ
)
ite
of
the
resulting
homomorphism
I
→
Aut(M
)
=
Aut(Π
G
M
Σ
)
→
Out(Π
G
Σ
)
is
an
outer
with
the
natural
projection
Aut(Π
G
M
M
representation
of
VA-type
(respectively,
NN-type;
PIPSC-
type)
in
the
sense
of
[NodNon],
Definition
2.4,
(ii)
[cf.
also
Remark
1.5.1
of
the
present
paper]
(respectively,
[NodNon],
Definition
2.4,
(iii);
Definition
1.3
of
the
present
paper).
Proof.
The
implication
(ii)
⇒
(i)
is
immediate;
the
implication
(i)
⇒
(ii)
follows
immediately
from
Lemma
1.5.
This
completes
the
proof
of
Lemma
1.9.
22
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Lemma
1.10
(Automorphisms
of
semi-graphs
of
anabelioids
of
PSC-type
with
prescribed
underlying
semi-graphs).
Let
Π
∗G
be
a
maximal
almost
pro-Σ
quotient
of
Π
G
[cf.
Definition
1.1]
and
α
∈
Out(Π
∗G
).
Suppose
that
there
exist
distinct
elements
v
1
,
v
2
,
v
3
∈
Vert(G);
e
1
,
e
2
∈
Node(G)
such
that
Vert(G)
=
{v
1
,
v
2
,
v
3
};
Node(G)
=
{e
1
,
e
2
};
V(e
i
)
=
{v
i
,
v
i+1
}
[where
i
∈
{1,
2}].
For
each
i
∈
{1,
2},
write
Π
G
∗
{e
}
for
the
maximal
almost
pro-Σ
quotient
of
i
Π
G
{ei
}
[cf.
[CbTpI],
Definition
2.8]
determined
by
the
natural
outer
∼
isomorphism
Φ
G
{ei
}
:
Π
G
{ei
}
→
Π
G
[cf.
[CbTpI],
Definition
2.10]
and
∼
the
maximal
almost
pro-Σ
quotient
Π
∗G
of
Π
G
;
Φ
∗G
{e
}
:
Π
∗G
{e
}
→
Π
∗G
i
i
for
the
outer
isomorphism
determined
by
Φ
G
{ei
}
.
Suppose,
moreover,
that,
for
each
i
∈
{1,
2},
the
outomorphism
of
Π
∗G
{e
}
obtained
by
con-
i
jugating
α
by
Φ
∗G
{e
}
is
a
profinite
Dehn
multi-twist
of
Π
∗G
{e
}
[cf.
i
i
Definition
1.6,
(v)].
Then
α
is
the
identity
outomorphism.
Proof.
First,
let
us
observe
that
it
follows
immediately
from
the
defi-
nition
of
a
profinite
Dehn
multi-twist
that
α
is
a
profinite
Dehn
multi-
twist
of
Π
∗G
.
Let
us
fix
a
verticial
subgroup
Π
v
2
⊆
Π
G
associated
to
v
2
.
Let
Π
e
1
,
Π
e
2
⊆
Π
G
be
nodal
subgroups
associated
to
e
1
,
e
2
,
respectively,
which
are
contained
in
Π
v
2
;
Π
v
1
⊆
Π
G
a
verticial
subgroup
associated
to
v
1
which
contains
Π
e
1
;
Π
v
3
⊆
Π
G
a
verticial
subgroup
associated
to
v
3
which
contains
Π
e
2
.
Thus,
we
have
inclusions
Π
v
1
⊇
Π
e
1
⊆
Π
v
2
⊇
Π
e
2
⊆
Π
v
3
.
For
each
z
∈
{v
1
,
v
2
,
v
3
,
e
1
,
e
2
},
write
Π
∗
z
⊆
Π
∗G
for
the
VCN-subgroup
of
Π
∗G
associated
to
z
obtained
by
forming
the
image
of
Π
z
⊆
Π
G
in
Π
∗G
.
Then
since
α
is
a
profinite
Dehn
multi-twist,
there
exists
a
lifting
α[v
2
]
∈
Aut(Π
∗G
)
of
α
which
preserves
and
induces
the
identity
auto-
morphism
of
Π
∗
v
2
;
in
particular,
α[v
2
]
preserves
and
induces
the
identity
automorphisms
of
Π
∗
e
1
,
Π
∗
e
2
.
Moreover,
by
applying
a
similar
argument
to
the
argument
given
in
the
proof
of
[CbTpI],
Lemma
4.6,
(i),
where
we
replace
[CmbGC],
Remark
1.1.3
(respectively,
[CmbGC],
Proposi-
tion
1.2,
(ii);
[CbTpI],
Proposition
4.5;
[NodNon],
Lemma
1.7),
in
the
proof
of
[CbTpI],
Lemma
4.6,
(i),
by
Proposition
1.7,
(ii)
(respectively,
Proposition
1.7,
(vii);
Remark
1.8.1;
Proposition
1.7,
(vi)),
we
conclude
that
α[v
2
](Π
∗
v
1
)
=
Π
∗
v
1
,
α[v
2
](Π
∗
v
3
)
=
Π
∗
v
3
,
and,
moreover,
that
there
ex-
ist
unique
elements
γ
1
∈
Π
∗
e
1
,
γ
2
∈
Π
∗
e
2
such
that
the
restrictions
of
α[v
2
]
to
Π
∗
v
1
,
Π
∗
v
3
are
the
inner
automorphisms
determined
by
γ
1
,
γ
2
,
respectively.
Thus,
since,
for
each
i
∈
{1,
2},
the
outomorphism
of
Π
∗G
{e
}
obtained
by
conjugating
α
by
Φ
∗G
{e
}
is
a
profinite
Dehn
multi-
i
i
twist
of
Π
∗G
{e
}
,
one
verifies
easily
—
by
considering
the
restriction
of
i
this
outomorphism
of
Π
∗G
{e
}
to
the
unique
conjugacy
class
of
verticial
i
subgroups
of
Π
∗G
{e
}
that
does
not
arise
from
a
conjugacy
class
of
ver-
i
ticial
subgroups
of
Π
∗G
[cf.
also
Proposition
1.7,
(ii),
(vii)]
—
that
γ
1
COMBINATORIAL
ANABELIAN
TOPICS
III
23
and
γ
2
are
trivial.
On
the
other
hand,
it
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
[CmbCsp],
Proposition
1.5,
(iii),
that
Π
G
is
topologically
generated
by
Π
v
1
,
Π
v
2
,
and
Π
v
3
;
in
particular,
Π
∗G
is
topologically
generated
by
Π
∗
v
1
,
Π
∗
v
2
,
and
Π
∗
v
3
.
Thus,
we
conclude
that
α[v
2
]
is
the
identity
automorphism
of
Π
∗G
.
This
completes
the
proof
of
Lemma
1.10.
Theorem
1.11
(Group-theoretic
verticiality/nodality
of
iso-
morphisms
of
outer
representations
of
NN-,
PIPSC-type).
Let
Σ
⊆
Σ
†
be
nonempty
sets
of
prime
numbers,
G
(respectively,
H)
a
semi-graph
of
anabelioids
of
pro-Σ
†
PSC-type,
Π
G
(respectively,
Π
H
)
the
[pro-Σ
†
]
fundamental
group
of
G
(respectively,
H),
Π
∗G
(respectively,
Π
∗H
)
a
maximal
almost
pro-Σ
quotient
[cf.
Definition
1.1]
of
Π
G
∼
(respectively,
Π
H
),
α
:
Π
∗G
→
Π
∗H
an
isomorphism
of
profinite
groups,
I
(respectively,
J)
a
profinite
group,
ρ
I
:
I
→
Out
grph
(Π
∗G
)
(respectively,
ρ
J
:
J
→
Out
grph
(Π
∗H
))
[cf.
Definition
1.6,
(iii)]
a
continuous
homo-
∼
morphism,
and
β
:
I
→
J
an
isomorphism
of
profinite
groups.
Suppose
that
the
diagram
ρ
I
→
Out(Π
∗G
)
I
−−−
⏐
⏐
⏐
⏐
Out(α)
β
ρ
J
J
−−−
→
Out(Π
∗H
)
—
where
the
right-hand
vertical
arrow
is
the
isomorphism
obtained
by
conjugating
by
α
—
commutes.
Then
the
following
hold:
(i)
Suppose,
moreover,
that
ρ
I
,
ρ
J
are
of
NN-type
[cf.
Defini-
tion
1.6,
(iv)].
Then
the
following
three
conditions
are
equiva-
lent:
(1)
The
isomorphism
α
is
group-theoretically
verticial
[i.e.,
roughly
speaking,
preserves
verticial
subgroups
—
cf.
Def-
inition
1.6,
(ii)].
(2)
The
isomorphism
α
is
group-theoretically
nodal
[i.e.,
roughly
speaking,
preserves
nodal
subgroups
—
cf.
Defini-
tion
1.6,
(ii)].
(3)
There
exists
an
infinite
subgroup
H
⊆
Π
∗G
of
Π
∗G
such
that
H
⊆
Π
∗G
,
α(H)
⊆
Π
∗H
are
contained
in
verticial
subgroups
of
Π
∗G
,
Π
∗H
,
respectively
[cf.
Definition
1.6,
(i)].
(ii)
Suppose,
moreover,
that
ρ
I
is
of
NN-type,
and
that
ρ
J
is
of
PIPSC-type
[cf.
Definition
1.6,
(iv)].
[For
example,
this
will
be
the
case
if
both
ρ
I
and
ρ
J
are
of
PIPSC-type
—
cf.
24
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Remark
1.6.2.]
Then
α
is
group-theoretically
verticial,
hence
also
group-theoretically
nodal.
Proof.
The
implication
(1)
⇒
(2)
of
assertion
(i)
and
the
final
portion
of
assertion
(ii)
[i.e.,
the
portion
concerning
group-theoretic
nodality]
fol-
low
immediately
from
Proposition
1.7,
(iv).
The
implication
(2)
⇒
(3)
of
assertion
(i)
is
immediate.
Finally,
we
verify
assertion
(ii)
(respec-
tively,
the
implication
(3)
⇒
(1)
of
assertion
(i)).
Suppose
that
ρ
I
,
ρ
J
are
as
in
assertion
(ii)
(respectively,
condition
(3)
of
assertion
(i)).
Let
N
G
⊆
Π
G
,
N
H
⊆
Π
H
be
normal
open
subgroups
of
Π
G
,
Π
H
with
respect
to
which
Π
∗G
,
Π
∗H
are
the
maximal
almost
pro-Σ
quotients
of
Π
G
,
Π
H
,
respectively.
[Thus,
N
G
Σ
⊆
Π
∗G
,
N
H
Σ
⊆
Π
∗H
.]
Now
it
follows
immediately
from
the
fact
that
Π
∗G
,
Π
∗H
are
topologically
finitely
gen-
erated
[cf.
Proposition
1.7,
(i)]
that
there
exists
a
characteristic
open
def
subgroup
M
G
⊆
Π
∗G
of
Π
∗G
such
that
M
G
⊆
N
G
Σ
,
M
H
=
α(M
G
)
⊆
N
H
Σ
.
Thus,
it
follows
immediately,
in
light
of
Lemma
1.9,
from
[CbTpII],
Theorem
1.9,
(ii)
(respectively,
the
implication
(3)
⇒
(1)
of
[CbTpII],
Theorem
1.9,
(i)),
together
with
Proposition
1.7,
(vii),
that
α
is
group-
theoretically
verticial.
This
completes
the
proof
of
Theorem
1.11.
Corollary
1.12
(Group-theoretic
graphicity
of
group-theoreti-
cally
cuspidal
isomorphisms
of
outer
representations
of
NN-,
PIPSC-type).
Let
Σ
⊆
Σ
†
be
nonempty
sets
of
prime
numbers,
G
(respectively,
H)
a
semi-graph
of
anabelioids
of
pro-Σ
†
PSC-type,
Π
G
(respectively,
Π
H
)
the
[pro-Σ
†
]
fundamental
group
of
G
(respectively,
H),
Π
∗G
(respectively,
Π
∗H
)
a
maximal
almost
pro-Σ
quotient
[cf.
∼
Definition
1.1]
of
Π
G
(respectively,
Π
H
),
α
:
Π
∗G
→
Π
∗H
an
isomor-
phism
of
profinite
groups,
I
(respectively,
J)
a
profinite
group,
ρ
I
:
I
→
Out
grph
(Π
∗G
)
(respectively,
ρ
J
:
J
→
Out
grph
(Π
∗H
))
[cf.
Definition
1.6,
∼
(iii)]
a
continuous
homomorphism,
and
β
:
I
→
J
an
isomorphism
of
profinite
groups.
Suppose
that
the
following
conditions
are
satisfied:
(i)
The
diagram
ρ
I
I
−−−
→
Out(Π
∗G
)
⏐
⏐
⏐
⏐
Out(α)
β
ρ
J
J
−−−
→
Out(Π
∗H
)
—
where
the
right-hand
vertical
arrow
is
the
isomorphism
ob-
tained
by
conjugating
by
α
—
commutes.
(ii)
α
is
group-theoretically
cuspidal
[cf.
Definition
1.6,
(ii)].
(iii)
ρ
I
,
ρ
J
are
of
NN-type
[cf.
Definition
1.6,
(iv)].
Suppose,
moreover,
that
one
of
the
following
conditions
is
satisfied:
COMBINATORIAL
ANABELIAN
TOPICS
III
25
(1)
Cusp(G)
=
∅.
(2)
Either
ρ
I
or
ρ
J
is
of
PIPSC-type
[cf.
Definition
1.6,
(iv)].
Then
α
is
group-theoretically
graphic
[cf.
Definition
1.6,
(iii)].
Proof.
This
follows
immediately
from
Theorem
1.11,
(i)
(respectively,
(ii)),
whenever
condition
(1)
(respectively,
(2))
is
satisfied.
26
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
2.
Almost
pro-Σ
injectivity
In
the
present
§2,
we
develop
an
almost
pro-Σ
version
of
the
injec-
tivity
portion
of
the
theory
of
combinatorial
cuspidalization
[cf.
The-
orem
2.9,
Corollary
2.10
below].
We
also
discuss
an
almost
pro-l
analogue
[cf.
Corollary
2.13
below]
of
the
tripod
homomorphism
of
[CbTpII],
Definition
3.19.
The
significance
of
developing
these
almost
pro-l
analogues
of
standard
results
in
combinatorial
anabelian
geome-
try
is
that
they
allow
one
to
apply
techniques
that
are,
a
priori,
only
available
in
the
pro-l
case
to
profinite
fundamental
groups.
Such
pro-l
techniques
will
be
necessary
to
verify
the
results
obtained
in
§3
concern-
ing
various
metric
aspects
of
the
outer
representations
of
Galois
groups
that
arise
from
hyperbolic
curves
and
their
associated
configuration
spaces
over
p-adic
fields.
In
the
present
§2,
let
Σ
be
a
nonempty
set
of
prime
numbers.
Definition
2.1.
Let
l
be
a
prime
number;
n
a
positive
integer;
(g,
r)
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
k
an
algebraically
closed
field
of
characteristic
zero;
(Spec
k)
log
the
log
scheme
obtained
by
equipping
Spec
k
with
the
log
structure
determined
by
the
fs
chart
N
→
k
that
maps
1
→
0;
X
log
a
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
.
For
each
positive
integer
i,
write
X
i
log
for
the
i-th
log
configuration
space
of
X
log
[cf.
the
discussion
entitled
“Curves”
in
[CbTpII],
§0];
Π
i
for
the
pro-Primes
configuration
space
group
[cf.
[ExtFam],
Theorem
B;
[MzTa],
Definition
2.3,
(i)]
given
by
the
kernel
of
the
natural
outer
surjection
π
1
(X
i
log
)
π
1
((Spec
k)
log
).
Let
Π
n
Π
n
∗
be
a
quotient
of
Π
n
.
Write
{1}
=
Π
n/n
⊆
Π
n/n−1
⊆
·
·
·
⊆
Π
n/m
⊆
·
·
·
⊆
Π
n/2
⊆
Π
n/1
⊆
Π
n/0
=
Π
n
for
the
standard
fiber
filtration
on
Π
n
—
i.e.,
Π
n/m
⊆
Π
n
is
the
kernel
of
some
fixed
surjection
[that
belongs
to
the
collection
of
surjections
that
constitutes
the
outer
surjection]
p
Π
n/m
:
Π
n
Π
m
induced
by
the
log
log
obtained
by
forgetting
the
factors
labeled
projection
p
n/m
:
X
n
log
→
X
m
by
indices
>
m
[cf.
[CmbCsp],
Definition
1.1,
(i)];
∗
⊆
·
·
·
⊆
Π
∗
n/2
⊆
Π
∗
n/1
⊆
Π
∗
n/0
=
Π
∗
n
{1}
=
Π
∗
n/n
⊆
Π
∗
n/n−1
⊆
·
·
·
⊆
Π
n/m
for
the
induced
filtration
on
Π
∗
n
.
(i)
For
each
1
≤
m
≤
n,
we
shall
refer
to
the
subquotient
Π
∗
n/m−1
/Π
∗
n/m
of
Π
∗
n
as
a
standard-adjacent
subquotient
of
Π
∗
n
.
(ii)
We
shall
say
that
Π
∗
n
is
an
SA-maximal
almost
pro-l
quo-
tient
of
Π
n
[where
the
“SA”
stands
for
“standard-adjacent”]
if,
for
every
1
≤
m
≤
n,
the
natural
quotient
Π
n/m−1
/Π
n/m
Π
∗
n/m−1
/Π
∗
n/m
is
a
maximal
almost
pro-l
quotient
of
Π
n/m−1
/Π
n/m
[cf.
Definition
1.1].
COMBINATORIAL
ANABELIAN
TOPICS
III
27
(iii)
We
shall
say
that
Π
∗
n
is
F-characteristic
if
every
F-admissible
automorphism
[cf.
[CmbCsp],
Definition
1.1,
(ii)]
of
Π
n
pre-
serves
the
kernel
of
the
quotient
Π
n
Π
∗
n
.
(iv)
We
shall
refer
to
the
image
of
a
fiber
subgroup
[cf.
[MzTa],
Definition
2.3,
(iii)]
of
Π
n
in
Π
∗
n
as
a
fiber
subgroup
of
Π
∗
n
.
For
each
1
≤
m
≤
n,
we
shall
refer
to
the
image
of
a
cuspidal
inertia
subgroup
of
Π
n/m−1
/Π
n/m
in
Π
∗
n/m−1
/Π
∗
n/m
as
a
cuspidal
inertia
subgroup
of
Π
∗
n/m−1
/Π
∗
n/m
.
(v)
Let
α
be
an
automorphism
of
Π
∗
n
.
Then
we
shall
say
that
α
is
F-admissible
if
α
preserves
each
fiber
subgroup
[cf.
(iv)]
of
Π
∗
n
.
We
shall
say
that
α
is
C-admissible
if
α
preserves
the
filtration
{1}
=
Π
∗
n/n
⊆
Π
∗
n/n−1
⊆
·
·
·
⊆
Π
∗
n/m
⊆
·
·
·
⊆
Π
∗
n/2
⊆
Π
∗
n/1
⊆
Π
∗
n/0
=
Π
∗
n
,
and,
moreover,
α
induces
a
bijection
of
the
set
of
cuspidal
iner-
tia
subgroups
[cf.
(iv)]
of
every
standard-adjacent
subquotient
[cf.
(i)]
of
Π
∗
n
.
We
shall
say
that
α
is
FC-admissible
if
α
is
F-admissible
and
C-admissible.
(vi)
Let
α
be
an
outomorphism
of
Π
∗
n
.
Then
we
shall
say
that
α
is
F-admissible
(respectively,
C-admissible;
FC-admissible)
if
α
arises
from
an
automorphism
of
Π
∗
n
that
is
F-admissible
(respectively,
C-admissible;
FC-admissible)
[cf.
(v)].
(vii)
Write
Aut
F
(Π
∗
n
)
,
Aut
C
(Π
∗
n
)
,
Aut
FC
(Π
∗
n
)
⊆
Aut(Π
∗
n
)
for
the
respective
subgroups
of
F-,
C-,
and
FC-admissible
au-
tomorphisms
of
Π
∗
n
[cf.
(v)];
Out
F
(Π
∗
n
)
=
Aut
F
(Π
∗
n
)/Inn(Π
∗
n
)
,
def
Out
C
(Π
∗
n
)
=
Aut
C
(Π
∗
n
)/Inn(Π
∗
n
)
,
def
Out
FC
(Π
∗
n
)
=
Aut
FC
(Π
∗
n
)/Inn(Π
∗
n
)
⊆
Out(Π
∗
n
)
for
the
respective
subgroups
of
F-,
C-,
and
FC-admissible
out-
omorphisms
of
Π
∗
n
[cf.
(vi)].
def
(viii)
Let
Π
n
Π
∗∗
n
be
a
quotient
of
Π
n
that
dominates
the
quotient
Π
n
Π
∗
n
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
Then
we
shall
write
F
∗
∗∗
Out
F
(Π
∗∗
n
Π
n
)
⊆
Out
(Π
n
)
,
FC
∗
∗∗
Out
FC
(Π
∗∗
n
Π
n
)
⊆
Out
(Π
n
)
[cf.
(vii)]
for
the
respective
subgroups
of
F-,
FC-admissible
outomorphisms
of
Π
∗∗
n
that
preserve
the
kernel
of
the
natural
∗
Π
.
surjection
Π
∗∗
n
n
Thus,
we
have
natural
homomorphisms
F
∗
∗
Out
F
(Π
∗∗
n
Π
n
)
−→
Out
(Π
n
)
,
28
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
FC
∗
∗
Out
FC
(Π
∗∗
n
Π
n
)
−→
Out
(Π
n
)
.
We
shall
write
F
∗
Out
F
(Π
∗
n
Π
∗∗
n
)
⊆
Out
(Π
n
)
,
FC
∗
Out
FC
(Π
∗
n
Π
∗∗
n
)
⊆
Out
(Π
n
)
for
the
respective
images
of
these
natural
homomorphisms.
Thus,
we
have
natural
surjections
F
∗
∗
∗∗
Out
F
(Π
∗∗
n
Π
n
)
Out
(Π
n
Π
n
)
,
FC
∗
∗
∗∗
Out
FC
(Π
∗∗
n
Π
n
)
Out
(Π
n
Π
n
)
.
Remark
2.1.1.
In
the
notation
of
Definition
2.1,
suppose
that
Π
∗
n
is
F-characteristic
[cf.
Definition
2.1,
(iii)].
Then
it
follows
from
the
various
definitions
involved
that
Out
F
(Π
n
Π
∗
n
)
=
Out
F
(Π
n
),
Out
FC
(Π
n
Π
∗
n
)
=
Out
FC
(Π
n
)
[cf.
Definition
2.1,
(viii)];
thus,
we
have
natural
surjections
Out
F
(Π
n
)
Out
F
(Π
∗
n
Π
n
),
Out
FC
(Π
n
)
Out
FC
(Π
∗
n
Π
n
)
[cf.
Definition
2.1,
(viii)].
Lemma
2.2
(Preservation
of
quotients
of
extensions).
Let
1
−−−→
N
−−−→
G
−−−→
Q
−−−→
1
⏐
⏐
⏐
⏐
⏐
⏐
1
−−−→
N
−−−→
G
−−−→
Q
−−−→
1
be
a
commutative
diagram
of
profinite
groups
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
surjective.
Write
G
∗
=
Ker(G
Q
Q)/Ker(N
N
)
def
and
N
∗
for
the
image
of
N
in
G
∗
.
Suppose
that
N
is
center-free.
Then
the
image
of
Ker(G
G)
in
G
∗
is
equal
to
the
centralizer
Z
G
∗
(N
∗
).
Proof.
Observe
that,
by
replacing
G
by
Ker(G
Q
Q)
(=
G
×
Q
Ker(Q
Q)),
we
may
assume
without
loss
of
generality
that
Q
=
{1}.
In
a
similar
vein,
by
replacing
G
by
G/Ker(N
N
),
we
may
assume
without
loss
of
generality
that
N
=
N
,
which
[since
Q
=
{1}]
implies
that
G
=
G
∗
,
N
=
N
∗
=
N
.
Then
one
verifies
easily
that
the
natural
inclusions
N
,
Ker(G
G)
→
G
determine
an
isomorphism
∼
N
×
Ker(G
G)
→
G.
Thus,
since
N
is
center-free,
we
obtain
that
Ker(G
G)
=
Z
G
(N
).
This
completes
the
proof
of
Lemma
2.2.
COMBINATORIAL
ANABELIAN
TOPICS
III
29
Proposition
2.3
(Existence
of
F-characteristic
SA-maximal
al-
most
pro-l
quotients).
In
the
notation
of
Definition
2.1,
let
Π
n
Π
∗
n
be
a
quotient
of
Π
n
.
Then
the
following
hold:
(i)
If
Π
∗
n
is
an
SA-maximal
almost
pro-l
quotient
of
Π
n
[cf.
Definition
2.1,
(ii)],
then
Π
∗
n
is
topologically
finitely
gen-
erated,
almost
pro-l
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0],
and
slim
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0].
(ii)
Let
0
≤
m
1
≤
m
2
≤
n
be
integers
and
(Π
n/m
1
/Π
n/m
2
)
‡
an
almost
pro-l
quotient
of
Π
n/m
1
/Π
n/m
2
.
Then
there
exists
an
F-characteristic
[cf.
Definition
2.1,
(iii)]
SA-maximal
almost
pro-l
quotient
Π
∗∗
n
of
Π
n
such
that
the
quotient
of
Π
n/m
1
/Π
n/m
2
determined
by
the
quotient
Π
n
Π
∗∗
n
domi-
nates
the
quotient
Π
n/m
1
/Π
n/m
2
(Π
n/m
1
/Π
n/m
2
)
‡
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
(iii)
Let
1
≤
m
≤
n
be
an
integer,
H
⊆
Π
n/m−1
/Π
n/m
a
VCN-
subgroup
of
Π
n/m−1
/Π
n/m
[cf.
[CbTpII],
Definition
3.1,
(iv)],
and
H
H
‡
an
almost
pro-l
quotient
of
H.
Then
there
exists
an
F-characteristic
SA-maximal
almost
pro-l
quo-
tient
Π
∗∗
n
of
Π
n
such
that
the
quotient
of
H
determined
by
the
‡
quotient
Π
n
Π
∗∗
n
dominates
the
quotient
H
H
.
Proof.
First,
we
verify
assertion
(i).
Observe
that
it
follows
imme-
diately
from
Proposition
1.7,
(i),
together
with
the
definition
of
an
SA-maximal
almost
pro-l
quotient,
that
Π
∗
n
is
a
successive
extension
of
almost
pro-l,
topologically
finitely
generated,
slim
profinite
groups.
Thus,
one
verifies
immediately
that
Π
∗
n
is
almost
pro-l,
topologically
finitely
generated,
and
slim.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
First,
observe
that
since
(Π
n/m
1
/Π
n/m
2
)
‡
may
be
regarded
as
an
almost
pro-l
quotient
of
Π
n/m
1
,
we
may
assume
def
without
loss
of
generality
that
m
2
=
n.
Write
m
=
m
1
.
If
m
=
n,
then
one
may
take
the
quotient
Π
∗∗
n
to
be
the
maximal
pro-l
quotient
of
Π
n
[cf.
[MzTa],
Proposition
2.2,
(i)].
Thus,
we
may
assume
without
loss
of
generality
that
m
≤
n
−
1.
Let
us
verify
assertion
(ii)
by
induction
on
n.
If
n
=
1,
then
as-
sertion
(ii)
follows
immediately
from
the
fact
that
Π
1
is
topologically
finitely
generated,
which
implies
that
the
topology
of
Π
1
admits
a
basis
of
characteristic
open
subgroups.
Thus,
we
suppose
that
n
≥
2,
and
that
the
induction
hypothesis
is
in
force.
Then
observe
that
since
the
subgroup
Π
n/n−1
⊆
Π
n
may
be
regarded
as
the
“Π
1
”
associated
to
some
stable
log
curve
of
type
(g,
r
+
n
−
1),
by
applying
the
induction
hy-
pothesis
to
the
quotient
Π
n/n−1
Π
‡
n/n−1
determined
by
the
quotient
Π
n/m
Π
‡
n/m
,
we
obtain
an
F-characteristic
SA-maximal
almost
pro-l
30
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
‡
quotient
Π
∗∗
n/n−1
of
Π
n/n−1
which
dominates
Π
n/n−1
Π
n/n−1
.
In
par-
ticular,
since
the
quotient
Π
n/n−1
Π
∗∗
n/n−1
is
F-characteristic,
hence
arises
from
a
subgroup
of
Π
n/n−1
which
is
normal
in
Π
n
,
we
thus
obtain
a
natural
outer
action
∼
Π
n
/Π
n/n−1
(
→
Π
n−1
)
→
Out(Π
∗∗
n/n−1
)
.
Since
the
profinite
group
Π
∗∗
n/n−1
is
almost
pro-l
and
topologically
finitely
generated
[cf.
assertion
(i)],
it
follows
immediately
that
the
outer
action
Π
n
/Π
n/n−1
→
Out(Π
∗∗
n/n−1
)
factors
through
an
almost
pro-l
quotient
Π
n
/Π
n/n−1
Q
of
Π
n
/Π
n/n−1
.
In
particular,
it
follows
that
the
natural
outer
action
Π
n/m
/Π
n/n−1
⊆
Π
n
/Π
n/n−1
→
Out(Π
∗∗
n/n−1
)
factors
through
an
al-
most
pro-l
quotient
of
Π
n/m
/Π
n/n−1
.
Note
that
this
implies
[cf.
the
slimness
of
Π
∗∗
n/n−1
proved
in
assertion
(i)]
that
there
exists
an
al-
most
pro-l
quotient
Π
n/m
Q
∗∗
of
Π
n/m
that
induces
the
quotient
Π
n/n−1
Π
∗∗
n/n−1
of
Π
n/n−1
.
Now
one
verifies
immediately
that
the
∗∗∗
quotient
Q
determined
by
the
intersection
of
the
kernels
of
the
two
quotients
Π
n/m
Π
‡
n/m
,
Π
n/m
Q
∗∗
is
an
almost
pro-l
quotient
of
Π
n/m
that
induces
the
quotient
Π
n/n−1
Π
∗∗
n/n−1
of
Π
n/n−1
.
Thus,
we
conclude
that
by
replacing
the
quotient
Π
n/m
Π
‡
n/m
by
this
quo-
tient
Q
∗∗∗
,
we
may
assume
without
loss
of
generality
that
the
quotient
Π
n/m
Π
‡
n/m
induces
the
quotient
Π
n/n−1
Π
∗∗
n/n−1
of
Π
n/n−1
.
Next,
let
us
observe
that
if
we
regard
Π
n
/Π
n/n−1
as
the
“Π
n−1
”
associated
to
some
stable
log
curve
of
type
(g,
r),
then:
•
If
we
apply
the
induction
hypothesis
to
the
almost
pro-l
quo-
tient
Π
n/m
/Π
n/n−1
Π
‡
n/m
/Π
‡
n/n−1
,
then
we
obtain
an
[F-
characteristic
SA-maximal]
almost
pro-l
quotient
Π
n
/Π
n/n−1
Q
‡
of
Π
n
/Π
n/n−1
which
induces
a
quotient
of
Π
n/m
/Π
n/n−1
that
dominates
the
quotient
Π
n/m
/Π
n/n−1
Π
‡
n/m
/Π
‡
n/n−1
.
•
If
we
apply
the
induction
hypothesis
to
any
almost
pro-l
quo-
tient
of
Π
n
/Π
n/n−1
that
dominates
both
Q
and
Q
‡
[e.g.,
the
quo-
tient
determined
by
the
intersection
of
the
kernels
determined
by
the
quotients
Q,
Q
‡
],
then
we
obtain
an
F-characteristic
SA-maximal
almost
pro-l
quotient
Π
n
/Π
n/n−1
(Π
n
/Π
n/n−1
)
∗∗
of
Π
n
/Π
n/n−1
that
dominates
Q
and,
moreover,
induces
a
quo-
tient
of
Π
n/m
/Π
n/n−1
that
dominates
Π
‡
n/m
/Π
‡
n/n−1
.
In
particu-
lar,
the
above
outer
action
Π
n
/Π
n/n−1
→
Out(Π
∗∗
n/n−1
)
factors
through
the
natural
surjection
Π
n
/Π
n/n−1
(Π
n
/Π
n/n−1
)
∗∗
.
COMBINATORIAL
ANABELIAN
TOPICS
III
def
31
out
∗∗
∗∗
Now
let
us
write
Π
∗∗
[cf.
the
discussion
n
=
Π
n/n−1
(Π
n
/Π
n/n−1
)
entitled
“Topological
groups”
in
[CbTpI],
§0
—
where
we
note
that
Π
∗∗
n/n−1
is
center-free
by
assertion
(i)].
Then
it
follows
immediately
from
Lemma
2.2
[which
allows
one
to
reduce
an
inclusion
assertion
concerning
“Ker(−)’s”
to
an
inclusion
assertion
concerning
centraliz-
ers]
and
the
various
definitions
involved,
together
with
our
assumption
that
the
quotient
Π
n/m
Π
‡
n/m
induces
the
quotient
Π
n/n−1
Π
∗∗
n/n−1
of
Π
n/n−1
,
that
Π
∗∗
n
is
an
SA-maximal
almost
pro-l
quotient
of
Π
n
such
that
the
quotient
of
Π
n/m
determined
by
Π
n
Π
∗∗
n
dominates
the
quo-
‡
tient
Π
n/m
Π
n/m
.
Finally,
it
follows
immediately
from
Lemma
2.2
[which
allows
one
to
reduce
an
F-characteristicity
assertion
concerning
“Ker(−)”
to
an
F-characteristicity
assertion
concerning
a
certain
cen-
tralizer],
together
with
the
fact
that
the
quotients
Π
n/n−1
Π
∗∗
n/n−1
and
Π
n
/Π
n/n−1
(Π
n
/Π
n/n−1
)
∗∗
are
F-characteristic,
that
Π
∗∗
n
is
F-
characteristic.
This
completes
the
proof
of
assertion
(ii).
Assertion
(iii)
follows
immediately
from
assertion
(ii),
together
with
Proposition
1.7,
(viii).
This
completes
the
proof
of
Proposition
2.3.
def
Definition
2.4.
In
the
notation
of
Definition
2.1,
write
Π
F
=
Π
2/1
,
def
def
Π
T
=
Π
2
,
Π
B
=
Π
1
;
thus,
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Π
F
−→
Π
T
−→
Π
B
−→
1
[cf.
the
notation
introduced
in
[CbTpI],
Definition
6.3].
Let
Π
F
Π
F
∗
be
a
maximal
almost
pro-Σ
quotient
of
Π
F
[cf.
Definition
1.1].
Then
we
shall
say
that
Π
F
Π
∗
F
is
base-admissible
if
the
kernel
of
Π
F
Π
∗
F
is
normal
in
Π
T
.
Thus,
if
Π
F
Π
∗
F
is
base-admissible,
then
the
quotient
Π
F
Π
∗
F
determines
a
quotient
Π
T
Π
∗
T
of
Π
T
which
fits
into
a
natural
commutative
diagram
of
profinite
groups
1
−−−→
Π
F
−−−→
Π
T
−−−→
Π
B
−−−→
1
⏐
⏐
⏐
⏐
1
−−−→
Π
∗
F
−−−→
Π
∗
T
−−−→
Π
B
−−−→
1
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
surjective.
Definition
2.5.
In
the
notation
of
Definition
2.4,
suppose
that
Π
F
Π
∗
F
is
base-admissible
[cf.
Definition
2.4];
thus,
we
have
a
quotient
Π
T
Π
∗
T
32
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
of
Π
T
that
fits
into
the
commutative
diagram
of
Definition
2.4.
Let
x
∈
X(k)
be
a
k-valued
point
of
the
underlying
scheme
X
of
X
log
.
(i)
We
shall
write
Π
G
x
Π
∗G
x
[cf.
[CbTpI],
Definition
6.3,
(i)]
for
the
maximal
almost
pro-Σ
quotient
of
Π
G
x
determined
by
the
quotient
Π
F
Π
∗
F
and
the
∼
isomorphism
Π
F
→
Π
G
x
fixed
in
[CbTpI],
Definition
6.3,
(i).
[Here,
we
note
that
this
quotient
Π
G
x
Π
∗G
x
is
independent
∼
of
the
choice
of
isomorphism
Π
F
→
Π
G
x
in
[CbTpI],
Definition
∼
6.3,
(i).]
Thus,
the
fixed
isomorphism
Π
F
→
Π
G
x
induces
an
∼
isomorphism
of
profinite
groups
Π
∗
F
→
Π
∗G
x
.
(ii)
For
c
∈
Cusp
F
(G)
[cf.
[CbTpI],
Definition
6.5,
(i)],
we
shall
refer
to
a
closed
subgroup
of
Π
∗
F
obtained
by
forming
the
im-
∼
age
—
via
the
isomorphism
Π
∗G
x
←
Π
∗
F
[cf.
(i)]
for
some
k-
valued
point
x
∈
X(k)
—
of
a
cuspidal
subgroup
of
Π
∗G
x
as-
sociated
to
the
cusp
of
G
x
corresponding
to
c
∈
Cusp
F
(G)
[cf.
[CbTpI],
Lemma
6.4,
(ii)]
as
a
cuspidal
subgroup
of
Π
∗
F
associ-
ated
to
c
∈
Cusp
F
(G).
Note
that
it
follows
immediately
from
[CbTpI],
Lemma
6.4,
(ii),
that
the
Π
∗
F
-conjugacy
class
of
a
cus-
pidal
subgroup
of
Π
∗
F
associated
to
c
∈
Cusp
F
(G)
depends
only
on
c
∈
Cusp
F
(G),
i.e.,
it
does
not
depend
on
the
choice
of
x
or
on
the
choices
of
isomorphisms
made
in
[CbTpI],
Definition
6.3,
(i).
(iii)
Recall
that
Π
T
=
Π
2
,
Π
F
=
Π
2/1
[cf.
Definition
2.4].
In
par-
ticular,
it
makes
sense
to
speak
of
F-/C-/FC-admissible
auto-
morphisms
or
outomorphisms
of
Π
∗
T
,
Π
∗
F
[cf.
Definition
2.1,
(v),
(vi)].
Lemma
2.6
(Maximal
almost
pro-Σ
quotients
of
VCN-sub-
groups).
In
the
notation
of
Definition
2.5,
let
Π
∗
c
F
⊆
Π
∗
F
be
a
cuspidal
diag
subgroup
of
Π
∗
F
[cf.
Definition
2.5,
(ii)]
associated
to
c
Fdiag
∈
Cusp
F
(G)
∗
⊆
Π
∗
F
for
the
normal
[cf.
[CbTpI],
Definition
6.5,
(i)].
Write
N
diag
∗
closed
subgroup
of
Π
F
topologically
normally
generated
by
Π
∗
c
F
⊆
Π
∗
F
.
diag
[Note
that
it
follows
immediately
from
[CbTpI],
Lemma
6.4,
(i),
(ii),
∗
is
normal
in
Π
∗
T
.]
Then
the
following
hold:
that
N
diag
∗
(i)
If
we
regard
Π
∗
F
/N
diag
as
a
quotient
of
Π
G
by
means
of
the
∼
natural
outer
isomorphism
Π
F
/N
diag
→
Π
G
of
[CbTpI],
Lemma
∗
,
then
6.6,
(i),
and
the
natural
surjection
Π
F
/N
diag
Π
∗
F
/N
diag
∗
∗
Π
F
/N
diag
is
a
maximal
almost
pro-Σ
quotient
of
Π
G
[cf.
Definition
1.1].
COMBINATORIAL
ANABELIAN
TOPICS
III
33
(ii)
Let
z
F
∈
VCN(G
x
),
Π
z
F
⊆
Π
G
x
a
VCN-subgroup
of
Π
G
x
asso-
ciated
to
z
F
,
and
Π
z
F
Π
z
‡
F
an
almost
pro-Σ
quotient
of
Π
z
F
.
Then
there
exists
a
base-admissible
[cf.
Definition
2.4]
maximal
almost
pro-Σ
quotient
Π
∗∗
F
of
Π
F
such
that
the
∗∗
quotient
Π
z
F
Π
z
F
determined
by
the
quotient
Π
F
Π
∗∗
F
‡
dominates
the
quotient
Π
z
F
Π
z
F
[cf.
the
discussion
enti-
tled
“Topological
groups”
in
§0].
(iii)
Let
z
F
∈
VCN(G
x
)
\
{c
Fdiag
}
and
Π
∗
z
F
⊆
Π
∗G
x
a
VCN-subgroup
of
Π
∗G
x
associated
to
z
F
[cf.
Definition
1.6,
(i)].
Suppose
that
either
•
z
F
∈
Edge(G
x
)
or
•
z
F
=
v
x
F
for
v
∈
Vert(G)
[cf.
[CbTpI],
Definition
6.3,
(ii)]
such
that
x
does
not
lie
on
v
[cf.
[CbTpI],
Definition
6.3,
(iii)].
Then
there
exist
a
maximal
almost
pro-Σ
quotient
Π
∗∗
F
of
∗∗
∗∗
⊆
Π
of
Π
associated
to
z
F
Π
F
and
a
VCN-subgroup
Π
∗∗
G
x
G
x
z
F
such
that
the
following
conditions
are
satisfied:
∗
(a)
Π
F
Π
∗∗
F
dominates
Π
F
Π
F
.
(b)
Π
F
Π
∗∗
F
is
base-admissible.
(c)
The
quotient
of
Π
∗∗
z
F
determined
by
the
composite
∼
∗∗
∗∗
∗
Π
∗∗
z
F
→
Π
G
x
←
Π
F
Π
F
factors
through
the
quotient
of
Π
∗∗
z
F
determined
by
the
com-
posite
∼
∗∗
∗∗
∗∗
∗∗
Π
∗∗
z
F
→
Π
G
x
←
Π
F
Π
F
/N
diag
∗∗
—
where
we
write
N
diag
for
the
normal
closed
subgroup
of
∗∗
Π
F
topologically
normally
generated
by
the
cuspidal
sub-
F
F
groups
of
Π
∗∗
F
associated
to
c
diag
∈
Cusp
(G).
Proof.
Assertion
(i)
follows
immediately
from
Lemma
1.2,
(i).
Asser-
tion
(ii)
follows
immediately
from
Proposition
1.7,
(viii),
together
with
Lemma
1.2,
(iii)
[cf.
also
Proposition
1.7,
(i)].
In
a
similar
vein,
as-
sertion
(iii)
follows
immediately,
in
light
of
the
injectivity
assertion
of
[CbTpI],
Lemma
6.6,
(iii),
from
Proposition
1.7,
(viii)
[applied
to
Π
F
/N
diag
],
together
with
Lemma
1.2,
(iii)
[cf.
also
Proposition
1.7,
(i)].
This
completes
the
proof
of
Lemma
2.6.
Lemma
2.7
(Outomorphisms
that
preserve
the
diagonal).
In
the
notation
of
Lemma
2.6,
let
α
∗
be
an
automorphism
of
Π
∗
T
over
Π
B
34
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[i.e.,
which
preserves
and
induces
the
identity
automorphism
on
the
quotient
Π
∗
T
Π
B
].
Write
α
F
∗
∈
Out(Π
∗
F
)
for
the
outomorphism
of
Π
∗
F
determined
by
of
α
∗
.
Then
the
following
hold:
(i)
Suppose
that
α
∗
preserves
Π
∗
c
F
diag
⊆
Π
∗
F
.
Then
the
automor-
∗
induced
by
α
∗
is
the
identity
automor-
phism
of
Π
∗
F
/N
diag
phism.
(ii)
Let
e
∈
Edge(G),
x
∈
X(k)
be
such
that
x
e
[cf.
[CbTpI],
Definition
6.3,
(iii)].
Suppose
that
α
F
∗
is
C-admissible
[cf.
Definition
2.5,
(iii)],
and
that
Edge(G)
=
{e}
∪
Cusp(G).
∼
Then
it
holds
that
α
F
∗
∈
Out
grph
(Π
∗G
x
)
(⊆
Out(Π
∗G
x
)
←
Out(Π
∗
F
))
[cf.
Definition
1.6,
(iii)].
If,
moreover,
α
∗
preserves
Π
∗
c
F
⊆
Π
∗
F
,
then
α
F
∗
∈
Out
tion
1.8].
|grph|
diag
(Π
∗G
x
)
(⊆
Out
grph
(Π
∗G
x
))
[cf.
Defini-
(iii)
If
α
∗
is
FC-admissible
[cf.
Definition
2.5,
(iii)],
then
α
∗
∗
∗
∗
preserves
the
Π
F
-conjugacy
class
of
Π
c
F
⊆
Π
F
.
diag
def
Proof.
First,
we
verify
assertion
(i).
Write
D
∗
=
N
Π
∗
T
(Π
∗
c
F
)
⊆
Π
∗
T
.
diag
Then
it
follows
immediately
from
Proposition
1.7,
(vii),
that
the
nat-
ural
inclusion
D
∗
→
Π
∗
T
fits
into
the
following
exact
sequence
1
−−−→
Π
∗
c
F
−−−→
D
∗
−−−→
Π
B
−−−→
1
⏐
diag
⏐
⏐
⏐
1
−−−→
Π
∗
F
−−−→
Π
∗
T
−−−→
Π
B
−−−→
1
—
where
the
horizontal
sequences
are
exact.
Thus,
assertion
(i)
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
the
first
assertion
of
[CbTpI],
Lemma
6.7,
(i)
[cf.
also
the
proof
of
[CmbCsp],
Proposition
1.2,
(iii)].
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
The
fact
that
α
F
∗
∈
Out
grph
(Π
∗G
x
)
∼
(⊆
Out(Π
∗G
x
)
←
Out(Π
∗
F
))
follows
immediately
from
Corollary
1.12,
to-
gether
with
a
similar
argument
to
the
argument
applied
in
the
proof
of
the
first
assertion
of
[CbTpI],
Lemma
6.7,
(ii).
Now
suppose,
moreover,
that
α
∗
preserves
Π
∗
c
F
⊆
Π
∗
F
.
Then
the
fact
that
α
F
∗
∈
Out
|grph|
(Π
∗G
x
)
diag
(⊆
Out
grph
(Π
∗G
x
))
follows
immediately
from
assertion
(i);
Lemma
2.6,
(i);
Proposition
1.7,
(iii),
(v),
together
with
a
similar
argument
to
the
argument
applied
in
the
proof
of
the
second
assertion
of
[CbTpI],
Lemma
6.7,
(ii).
This
completes
the
proof
of
assertion
(ii).
Finally,
assertion
(iii)
follows
immediately,
in
light
of
Lemma
2.6,
(i),
from
the
definition
of
FC-admissibility
[cf.
also
Proposition
1.7,
(v)].
This
completes
the
proof
of
Lemma
2.7.
COMBINATORIAL
ANABELIAN
TOPICS
III
35
Lemma
2.8
(Triviality
of
certain
outomorphisms).
In
the
nota-
tion
of
Definition
2.5,
there
exists
a
base-admissible
maximal
al-
most
pro-Σ
quotient
Π
F
Π
∗∗
F
[cf.
Definitions
1.1;
2.4]
of
Π
F
that
∗
dominates
Π
F
Π
F
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
such
that
the
following
condition
(‡)
is
satisfied:
∗
(‡):
Let
α
∗
be
an
automorphism
of
Π
T
.
Then
for
any
base-admissible
maximal
almost
pro-Σ
quotient
Π
F
Π
∗∗∗
that
dominates
Π
F
Π
∗∗
∗
arises
F
F
,
if
α
from
an
FC-admissible
automorphism
[cf.
Defini-
tion
2.5,
(iii)]
of
Π
∗∗∗
[where
we
write
Π
∗∗∗
for
the
T
T
∗
∗∗∗
∗∗∗
∗∗∗
∼
“Π
T
”
determined
by
Π
F
]
over
Π
T
/Π
F
→
Π
B
,
then
α
∗
is
Π
∗
F
-inner.
Proof.
The
following
argument
is
essentially
the
same
as
the
argument
applied
in
[CmbCsp],
[NodNon],
[CbTpI]
to
prove
[CmbCsp],
Corollary
2.3,
(ii);
[NodNon],
Corollary
5.3;
[CbTpI],
Lemma
6.8,
respectively.
Let
us
fix
a
cuspidal
subgroup
Π
∗
c
F
⊆
Π
∗
F
of
Π
∗
F
[cf.
Definition
2.5,
diag
(ii)]
associated
to
c
Fdiag
∈
Cusp
F
(G)
[cf.
[CbTpI],
Definition
6.5,
(i)].
Let
Π
F
Π
∗∗
F
be
a
base-admissible
maximal
almost
pro-Σ
quotient
of
Π
F
that
dominates
Π
F
Π
∗
F
;
Π
F
Π
∗∗∗
a
base-admissible
max-
F
∗
an
imal
almost
pro-Σ
quotient
of
Π
F
that
dominates
Π
F
Π
∗∗
F
;
α
automorphism
of
Π
∗
T
that
arises
from
an
FC-admissible
automorphism
∗∗∗
∼
α
∗∗∗
of
Π
∗∗∗
over
Π
∗∗∗
→
Π
B
.
Here,
let
us
observe
that
one
T
T
/Π
F
∗
verifies
easily
that
α
is
an
FC-admissible
automorphism
of
Π
∗
T
over
∼
∗
.
Π
∗
T
/Π
∗
F
→
Π
B
.
Write
α
F
∗
for
the
outomorphism
of
Π
∗
F
determined
by
α
∗
∗
∗
Observe
that
since
α
F
preserves
the
Π
F
-conjugacy
class
of
Π
c
F
⊆
Π
∗
F
diag
[cf.
Lemma
2.7,
(iii)],
we
may
assume
without
loss
of
generality
—
by
∗∗∗
—
that
α
∗
preserves
replacing
α
∗∗∗
by
a
suitable
Π
∗∗∗
F
-conjugate
of
α
Π
∗
c
F
⊆
Π
∗
F
,
and
hence
[cf.
Lemma
2.7,
(i),
(ii)]
that
diag
∗
(a)
the
automorphism
of
Π
∗
F
/N
diag
induced
by
α
∗
is
the
identity
automorphism;
(b)
for
e
∈
Edge(G),
x
∈
X(k)
such
that
x
e
[cf.
[CbTpI],
Definition
6.3,
(iii)],
if
Edge(G)
=
{e}
∪
Cusp(G),
then
α
F
∗
∈
∼
Out
|grph|
(Π
∗G
x
)
(⊆
Out(Π
∗G
x
)
←
Out(Π
∗
F
))
[cf.
Definition
1.8].
Now
we
claim
that
the
following
assertion
holds:
Claim
2.8.A:
Lemma
2.8
holds
if
(g,
r)
=
(0,
3).
Indeed,
write
c
1
,
c
2
,
c
3
∈
Cusp(G)
for
the
three
distinct
cusps
of
G;
v
∈
Vert(G)
for
the
unique
vertex
of
G.
For
i
∈
{1,
2,
3},
let
x
i
∈
X(k)
be
such
that
x
i
c
i
.
Next,
let
us
observe
that
since
our
assumption
that
(g,
r)
=
(0,
3)
implies
that
Node(G)
=
∅,
it
follows
immediately
∼
from
(b)
that,
for
i
∈
{1,
2,
3},
the
outomorphism
α
F
∗
of
Π
∗G
xi
←
Π
∗
F
36
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∼
is
∈
Out
|grph|
(Π
∗G
xi
)
(⊆
Out(Π
∗G
xi
)
←
Out(Π
∗
F
)).
Next,
let
us
fix
a
ver-
∼
ticial
subgroup
Π
∗
v
x
F
⊆
Π
∗G
x
2
←
Π
∗
F
associated
to
v
x
F
2
∈
Vert(G
x
2
)
[cf.
2
[CbTpI],
Definition
6.3,
(ii)].
Then
since
α
F
∗
∈
Out
|grph|
(Π
∗G
x
2
),
it
fol-
lows
immediately
from
the
[easily
verified]
surjectivity
of
the
composite
∼
∗
∗
Π
∗
v
x
F
→
Π
∗G
x
2
←
Π
∗
F
Π
∗
F
/N
diag
that
there
exists
an
N
diag
-conjugate
2
β
∗
of
α
∗
such
that
β
∗
(Π
∗
F
)
=
Π
∗
F
.
Thus,
it
follows
immediately
from
v
x
2
v
x
2
Lemma
2.6,
(iii)
—
by
replacing
Π
∗∗
F
by
a
suitable
base-admissible
max-
imal
almost
pro-Σ
quotient
Π
F
Π
∗∗
F
[i.e.,
a
quotient
as
in
Lemma
2.6,
(iii)]
that
dominates
the
original
Π
F
Π
∗∗
F
and
applying
the
conclusion
∗
∗
∗
“
β
(Π
v
x
F
)
=
Π
v
x
F
”,
together
with
the
property
(a)
discussed
above,
in
2
2
∗∗∗
∈
Aut(Π
∗∗∗
the
case
where
“
α
∗
”
is
taken
to
be
α
T
)
—
that
we
may
assume
without
loss
of
generality
that
(‡
1
):
β
∗
fixes
and
induces
the
identity
automorphism
∼
on
Π
∗
v
x
F
⊆
Π
∗G
x
2
←
Π
∗
F
.
2
Next,
let
Π
∗
c
F
1
⊆
Π
∗
F
be
a
cuspidal
subgroup
of
Π
∗
F
associated
to
c
F1
∈
Cusp
F
(G)
[cf.
[CbTpI],
Definition
6.5,
(i)]
that
is
contained
in
∼
∼
Π
∗
v
x
F
⊆
Π
∗G
x
2
←
Π
∗
F
;
Π
∗
v
x
F
⊆
Π
∗G
x
3
←
Π
∗
F
a
verticial
subgroup
associated
2
3
to
v
x
F
3
∈
Vert(G
x
3
)
that
contains
Π
∗
c
F
⊆
Π
∗
F
.
Then
it
follows
from
the
1
inclusion
Π
∗
F
⊆
Π
∗
F
,
together
with
(‡
),
that
β
∗
(Π
∗
F
)
=
Π
∗
F
.
Thus,
v
x
2
c
1
1
c
1
c
1
∼
∗
∗
since
the
verticial
subgroup
Π
v
x
F
⊆
Π
G
x
3
←
Π
∗
F
is
the
unique
verticial
3
∼
subgroup
of
Π
∗G
x
3
←
Π
∗
F
associated
to
v
x
F
3
∈
Vert(G
x
3
)
that
contains
Π
∗
c
F
[cf.
Proposition
1.7,
(v),
(vi)],
it
follows
immediately
from
the
1
fact
that
α
F
∗
∈
Out
|grph|
(Π
∗G
x
3
)
that
β
∗
(Π
∗
v
x
F
)
=
Π
∗
v
x
F
.
In
particular,
3
3
it
follows
immediately
from
Lemma
2.6,
(iii)
—
by
replacing
Π
∗∗
F
by
a
suitable
base-admissible
maximal
almost
pro-Σ
quotient
Π
F
Π
∗∗
F
[i.e.,
a
quotient
as
in
Lemma
2.6,
(iii)]
that
dominates
the
original
∗
∗
∗
Π
F
Π
∗∗
F
and
applying
the
conclusion
“
β
(Π
v
x
F
)
=
Π
v
x
F
”,
together
3
3
with
the
property
(a)
discussed
above,
in
the
case
where
“
α
∗
”
is
taken
to
be
α
∗∗∗
∈
Aut(Π
∗∗∗
T
)
—
that
we
may
assume
without
loss
of
generality
that
(‡
2
):
β
∗
fixes
and
induces
the
identity
automorphism
∼
on
Π
∗
v
x
F
⊆
Π
∗G
x
3
←
Π
∗
F
.
3
On
the
other
hand,
since
Π
∗
F
is
topologically
generated
by
Π
∗
v
x
F
⊆
∼
∼
2
Π
∗G
x
2
←
Π
∗
F
and
Π
∗
v
x
F
⊆
Π
∗G
x
3
←
Π
∗
F
[cf.
[CmbCsp],
Lemma
1.13],
(‡
1
)
3
and
(‡
2
)
imply
that
β
∗
induces
the
identity
automorphism
on
Π
∗
F
.
This
completes
the
proof
of
Claim
2.8.A.
Next,
we
claim
that
the
following
assertion
holds:
COMBINATORIAL
ANABELIAN
TOPICS
III
37
Claim
2.8.B:
Lemma
2.8
holds
if
(g,
r)
=
(1,
1).
Indeed,
let
us
first
observe
that
by
working
with
2-cuspidalizable
de-
generation
structures
[cf.
[CbTpII],
Definition
3.23,
(i),
(v)]
that
arise
scheme-theoretically
via
a
specialization
isomorphism
as
in
the
discus-
sion
preceding
[CmbCsp],
Definition
2.1
[cf.
also
[CbTpI],
Remark
5.6.1],
we
may
switch
back
and
forth,
at
will,
between
the
case
of
smooth
and
non-smooth
“X
log
”.
In
particular,
we
may
assume
without
loss
of
generality
that
(Vert(G)
,
Cusp(G)
,
Node(G)
)
=
(1,
1,
1).
Let
v
be
the
unique
vertex
of
G,
c
the
unique
cusp
of
G,
e
the
unique
node
of
G,
x
∈
X(k)
such
that
x
c
[cf.
[CbTpI],
Definition
6.3,
(iii)],
and
H
the
sub-semi-graph
of
PSC-type
[cf.
[CbTpI],
Definition
2.2,
(i)]
of
the
underlying
semi-graph
G
x
of
G
x
whose
set
of
vertices
=
{v
x
F
}
[cf.
[CbTpI],
Definition
6.3,
(ii)].
Then
it
follows
from
[CbTpI],
Lemma
6.4,
(iv),
that
there
exists
a
unique
node
e
Fnew,x
of
G
x
such
F
)
[cf.
[CbTpI],
Lemma
6.4,
(iii)].
Thus,
one
that
e
Fnew,x
∈
N
(v
new,x
verifies
easily
that
there
exists
a
unique
element
e
F
x
∈
N
(v
x
F
)
such
that
N
(v
x
F
)
=
{e
Fnew,x
,
e
F
x
}.
Let
us
fix
∼
•
a
nodal
subgroup
Π
∗
e
Fnew,x
⊆
Π
∗G
x
←
Π
∗
F
associated
to
e
Fnew,x
[cf.
Figure
1
below].
H
◦
•
e
Fnew,x
F
v
new,x
v
x
F
•
e
F
x
◦
Π
∗
sub
F
Figure
1:
G
x
Then
it
follows
immediately
—
by
applying
Proposition
1.7,
(v),
(vi),
in
the
situation
that
arises
in
the
case
of
a
smooth
“X
log
”
of
type
(1,
1)
[cf.
the
observations
made
above
concerning
degeneration
structures]
—
that
there
exist
∼
⊆
Π
∗G
x
←
Π
∗
F
associated
to
•
a
unique
verticial
subgroup
Π
∗
v
new,x
F
F
v
new,x
and
38
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∼
•
a
unique
subgroup
Π
∗
(G
x
)|
H
⊆
Π
∗G
x
←
Π
∗
F
that
belongs
to
the
Π
∗
F
-conjugacy
class
of
subgroups
that
arises
as
the
image
of
the
natural
outer
homomorphism
Π
(G
x
)|
H
→
Π
G
x
Π
∗G
x
[cf.
[CbTpI],
Definition
2.2,
(ii)]
such
that
Π
∗
e
Fnew,x
⊆
Π
∗
v
new,x
,
Π
∗
(G
x
)|
H
.
Moreover,
one
verifies
easily
—
F
by
applying
the
property
(b)
discussed
above
in
the
situation
that
arises
in
the
case
of
a
smooth
“X
log
”
of
type
(1,
1)
[cf.
the
observa-
tions
made
above
concerning
degeneration
structures]
—
that
α
F
∗
pre-
∼
serves
the
Π
∗
F
-conjugacy
classes
of
Π
∗
e
Fnew,x
,
Π
∗
v
new,x
,
Π
∗
(G
x
)|
H
⊆
Π
∗G
x
←
Π
∗
F
.
F
Thus,
it
follows
immediately
from
the
commensurable
terminality
of
∼
∗
[cf.
the
image
of
the
composite
Π
∗
e
Fnew,x
→
Π
∗G
x
←
Π
∗
F
Π
∗
F
/N
diag
Proposition
1.7,
(vii);
Lemma
2.6,
(i)],
together
with
the
property
(a)
∗
-conjugate
β
∗
of
α
∗
such
discussed
above,
that
there
exists
an
N
diag
that
β
∗
(Π
∗
e
Fnew,x
)
=
Π
∗
e
Fnew,x
.
In
particular,
in
light
of
the
uniqueness
and
Π
∗
(G
x
)|
H
,
properties
applied
above
to
specify
the
subgroups
Π
∗
v
new,x
F
we
conclude
that
β
∗
(Π
∗
F
)
=
Π
∗
F
,
β
∗
(Π
∗
)
=
Π
∗
.
Thus,
v
new,x
v
new,x
(G
x
)|
H
(G
x
)|
H
it
follows
immediately
from
Lemma
2.6,
(iii)
—
by
replacing
Π
∗∗
F
by
a
suitable
base-admissible
maximal
almost
pro-Σ
quotient
Π
F
Π
∗∗
F
[i.e.,
a
quotient
as
in
Lemma
2.6,
(iii),
applied
in
the
situation
that
arises
in
the
case
of
a
smooth
“X
log
”
of
type
(1,
1)
—
cf.
the
observations
made
above
concerning
degeneration
structures]
that
dominates
the
original
∗
∗
∗
Π
F
Π
∗∗
F
and
applying
the
conclusion
“
β
(Π
(G
x
)|
H
)
=
Π
(G
x
)|
H
”,
to-
gether
with
the
property
(a)
discussed
above,
in
the
case
where
“
α
∗
”
∗∗∗
∗∗∗
is
taken
to
be
α
∈
Aut(Π
T
)
—
that
we
may
assume
without
loss
of
generality
that
(‡
3
):
β
∗
fixes
and
induces
the
identity
automorphism
∼
on
Π
∗
(G
x
)|
H
⊆
Π
∗G
x
←
Π
∗
F
.
Next,
let
us
write
∼
•
Π
∗
v
x
F
⊆
Π
∗G
x
←
Π
∗
F
for
the
unique
[cf.
Proposition
1.7,
(v),
(vi)]
verticial
subgroup
associated
to
v
x
F
[cf.
[CbTpI],
Definition
6.3,
(ii)]
such
that
Π
∗
e
Fnew,x
⊆
Π
∗
v
x
F
⊆
Π
∗
(G
x
)|
H
.
[Note
that
it
follows
immediately
from
the
various
definitions
involved
that
such
a
verticial
subgroup
associated
to
v
x
F
always
exists.]
Then
since
Π
∗
v
x
F
⊆
Π
∗
(G
x
)|
H
,
it
fol-
lows
from
(‡
3
)
that
β
∗
fixes
and
induces
the
identity
automorphism
∼
[cf.
the
discus-
on
Π
∗
F
⊆
Π
∗
←
Π
∗
.
Thus,
since
β
∗
(Π
∗
F
)
=
Π
∗
F
v
x
G
x
F
v
new,x
v
new,x
sion
preceding
(‡
3
)],
we
conclude
that
β
∗
preserves
the
closed
subgroup
COMBINATORIAL
ANABELIAN
TOPICS
III
39
Π
∗
F
sub
⊆
Π
∗
F
of
Π
∗
F
obtained
by
forming
the
image
of
the
natural
homo-
morphism
lim
Π
∗
v
new,x
←
Π
∗
e
Fnew,x
→
Π
∗
v
x
F
−→
Π
∗
F
F
−→
—
where
the
inductive
limit
is
taken
in
the
category
of
profinite
groups.
Next,
let
us
observe
that
the
Π
∗
F
-conjugacy
class
of
Π
∗
F
sub
⊆
Π
∗
F
coin-
cides
with
the
Π
∗
F
-conjugacy
class
of
the
image
Π
∗
(G
x
)
F
[cf.
[CbTpI],
{e
x
}
Definition
2.5,
(ii)]
of
the
composite
∼
Π
(G
x
)
{e
F
}
→
Π
G
x
←
Π
F
Π
∗
F
x
—
where
the
first
arrow
is
the
natural
outer
injection
discussed
in
[CbTpI],
Proposition
2.11,
and
we
recall
that
e
F
x
is
the
node
of
G
x
that
corresponds
to
the
node
e
of
G.
On
the
other
hand,
if
we
write
for
the
maximal
almost
pro-Σ
quotient
of
Π
(G
x
)
{e
F
}
[cf.
Π
∗
(G
x
)
F
{e
new
}
new
[CbTpI],
Definition
2.8]
determined
by
the
maximal
almost
pro-Σ
quo-
∼
tient
Π
∗G
x
and
the
natural
outer
isomorphism
Φ
(G
x
)
{e
F
}
:
Π
(G
x
)
{e
F
}
→
new
new
Π
G
x
[cf.
[CbTpI],
Definition
2.10],
then
Π
∗
F
sub
may
be
regarded
as
a
ver-
∼
∼
ticial
subgroup
of
Π
∗
(G
x
)
F
→
Π
∗G
x
←
Π
∗
F
[cf.
[CbTpI],
Proposition
{e
new
}
2.9,
(i),
(3)].
Thus,
it
follows
from
Proposition
1.7,
(vii),
that
Π
∗
F
sub
is
commensurably
terminal
in
Π
∗
F
.
Next,
let
us
observe
that,
by
applying
a
similar
argument
to
the
ar-
gument
given
in
[CmbCsp],
Definition
2.1,
(iii),
(vi),
or
[NodNon],
Def-
inition
5.1,
(ix),
(x)
[i.e.,
roughly
speaking,
by
considering
the
portion
of
the
underlying
scheme
X
2
of
X
2
log
corresponding
to
the
underlying
scheme
(X
v
)
2
of
the
2-nd
log
configuration
space
(X
v
)
log
2
of
the
stable
log
log
curve
X
v
determined
by
G|
v
—
cf.
[CbTpI],
Definition
2.1,
(iii)],
∼
one
concludes
that
there
exists
a
verticial
subgroup
Π
v
⊆
Π
G
←
Π
B
associated
to
v
∈
Vert(G)
such
that
the
outer
action
of
Π
v
on
Π
∗
F
de-
ρ
∗
2/1
termined
by
the
composite
Π
v
→
Π
B
→
Out(Π
∗
F
)
—
where
we
write
ρ
∗
2/1
for
the
outer
action
determined
by
the
exact
sequence
of
profinite
groups
1
−→
Π
∗
F
−→
Π
∗
T
−→
Π
B
−→
1
—
preserves
the
Π
∗
F
-conjugacy
class
of
the
commensurably
terminal
subgroup
Π
∗
F
sub
⊆
Π
∗
F
[so
we
obtain
a
natural
outer
representation
Π
v
→
Out(Π
∗
F
sub
)
—
cf.
[CbTpI],
Lemma
2.12,
(iii)],
and,
moreover,
def
out
that
if
we
write
Π
∗
T
sub
=
Π
∗
F
sub
Π
v
(⊆
Π
∗
T
)
[cf.
the
discussion
enti-
tled
“Topological
groups”
in
[CbTpI],
§0],
then
it
follows
from
Propo-
sition
1.7,
(ii),
that
Π
∗
T
sub
is
naturally
isomorphic
to
a
profinite
group
of
the
form
“Π
∗
T
”
obtained
by
taking
“G”
to
be
G|
v
.
Now
since
β
∗
(Π
∗
F
sub
)
=
Π
∗
F
sub
,
and
α
∗
is
an
automorphism
over
the
∼
quotient
Π
∗
F
/Π
∗
T
→
Π
B
,
one
verifies
immediately
that
β
∗
determines
an
∗
automorphism
β
T
∗
sub
of
Π
T
sub
over
Π
v
.
Thus,
since
G|
v
is
of
type
(0,
3)
40
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[cf.
[CbTpI],
Definition
2.3,
(i)],
by
considering
a
diagram
similar
to
the
diagram
of
[CmbCsp],
Definition
2.1,
(vi),
or
[NodNon],
Definition
5.1,
(x),
and
applying
Claim
2.8.A
[cf.
also
Lemma
2.6,
(ii)],
we
conclude
—
by
replacing
Π
∗∗
F
by
a
suitable
base-admissible
maximal
almost
pro-Σ
∗∗
quotient
Π
F
Π
∗∗
F
that
dominates
the
original
Π
F
Π
F
and
applying
the
conclusion
that
β
∗
determines
an
automorphism
of
Π
∗
T
sub
over
Π
v
in
the
case
where
“
α
∗
”
is
taken
to
be
α
∗∗∗
∈
Aut(Π
∗∗∗
T
)
—
that
we
may
assume
without
loss
of
generality
that
(‡
4
):
β
T
∗
sub
is
a
Π
∗
F
sub
-inner
automorphism.
Moreover,
since
β
∗
fixes
and
induces
the
identity
automorphism
on
Π
∗
v
x
F
[cf.
the
discussion
following
(‡
3
)],
and
Π
∗
v
x
F
is
commensurably
ter-
minal
in
[Π
∗
F
,
hence
also
in]
Π
∗
F
sub
[cf.
Proposition
1.7,
(vii)]
and
slim
[cf.
Proposition
1.7,
(ii)],
we
conclude
that
β
T
∗
sub
is
the
identity
auto-
morphism;
in
particular,
since
Π
∗
v
new,x
⊆
Π
∗
F
sub
,
β
∗
induces
the
identity
F
.
Thus,
since
Π
∗
F
is
topologically
generated
by
automorphism
on
Π
∗
v
new,x
F
[cf.
[CmbCsp],
Proposition
2.2,
(iii)],
it
follows
from
Π
∗
(G
x
)|
H
and
Π
∗
v
new,x
F
(‡
3
)
that
β
∗
is
the
identity
automorphism.
This
completes
the
proof
of
Claim
2.8.B.
Finally,
we
claim
that
the
following
assertion
holds:
Claim
2.8.C:
Lemma
2.8
holds
for
arbitrary
(g,
r).
We
verify
Claim
2.8.C
by
induction
on
3g
−
3
+
r.
If
3g
−
3
+
r
=
0,
i.e.,
(g,
r)
=
(0,
3),
then
Claim
2.8.C
amounts
to
Claim
2.8.A.
On
the
other
hand,
if
(g,
r)
=
(1,
1),
then
Claim
2.8.C
amounts
to
Claim
2.8.B.
Thus,
we
suppose
that
3g
−
3
+
r
>
0,
that
(g,
r)
=
(1,
1),
and
that
the
induction
hypothesis
is
in
force.
Since
3g
−
3
+
r
>
0
and
(g,
r)
=
(1,
1),
one
verifies
easily
that
there
exists
a
stable
log
curve
Y
log
of
type
(g,
r)
over
(Spec
k)
log
such
that
Y
log
has
precisely
one
node
and
precisely
two
vertices.
Thus,
by
working
with
2-cuspidalizable
de-
generation
structures
[cf.
[CbTpII],
Definition
3.23,
(i),
(v)]
that
arise
scheme-theoretically
via
a
specialization
isomorphism
as
in
the
discus-
sion
preceding
[CmbCsp],
Definition
2.1
[cf.
also
[CbTpI],
Remark
5.6.1],
we
may
replace
X
log
by
Y
log
and
assume
without
loss
of
gener-
ality
that
(Vert(G)
,
Node(G)
)
=
(2,
1).
Let
e
be
the
unique
node
of
G
and
x
∈
X(k)
such
that
x
e
[cf.
[CbTpI],
Definition
6.3,
(iii)].
Next,
let
us
observe
that
since
Node(G)
=
{e}
=
1,
it
follows
from
the
property
(b)
discussed
above
∼
that
α
F
∗
∈
Out
|grph|
(Π
∗G
x
)
(⊆
Out(Π
∗G
x
)
←
Out(Π
∗
F
)).
Write
{e
F1
,
e
F2
}
=
F
N
(v
new,x
)
[cf.
[CbTpI],
Lemma
6.4,
(iv)].
Also,
for
i
∈
{1,
2},
denote
by
v
i
∈
Vert(G)
the
vertex
of
G
such
that
(v
i
)
F
x
∈
Vert(G
x
)
[cf.
[CbTpI],
F
Definition
6.3,
(ii)]
is
the
unique
element
of
V(e
F
i
)\{v
new,x
}
[cf.
[CbTpI],
Lemma
6.4,
(iv)];
by
H
i
the
sub-semi-graph
of
PSC-type
[cf.
[CbTpI],
COMBINATORIAL
ANABELIAN
TOPICS
III
41
Definition
2.2,
(i)]
of
the
underlying
semi-graph
G
x
of
G
x
whose
set
of
F
,
(v
i
)
F
x
}
[cf.
Figure
2
below].
vertices
=
{v
new,x
∼
For
i
∈
{1,
2},
let
Π
∗
(v
i
)
F
x
⊆
Π
∗G
x
←
Π
∗
F
be
a
verticial
subgroup
of
∼
F
}.
Then
since
Π
∗G
x
←
Π
∗
F
associated
to
the
vertex
(v
i
)
F
x
∈
V(e
F
i
)
\
{v
new,x
|grph|
∗
∗
∗
∗
(Π
G
x
),
it
follows
that
α
preserves
the
Π
F
-conjugacy
class
α
F
∈
Out
∼
∗
∗
∗
of
Π
(v
i
)
F
x
⊆
Π
G
x
←
Π
F
.
Thus,
since
the
image
of
the
composite
H
1
◦
·
·
·
·
·
·
·
◦
F
v
new,x
(v
1
)
F
x
•
•
e
F1
◦
(v
2
)
F
x
e
F2
•
◦
·
·
·
·
·
·
·
◦
H
2
Figure
2:
G
x
∗
Π
∗
(v
i
)
F
x
→
Π
∗
F
Π
∗
F
/N
diag
is
commensurably
terminal
[cf.
Proposi-
tion
1.7,
(vii);
Lemma
2.6,
(i)],
it
follows
immediately
from
the
property
∗
(a)
discussed
above
that
there
exists
an
N
diag
-conjugate
β
i
∗
[which
may
depend
on
i
∈
{1,
2}!]
of
α
∗
such
that
β
i
∗
(Π
∗
(v
i
)
F
x
)
=
Π
∗
(v
i
)
F
x
.
Therefore,
it
follows
immediately
from
Lemma
2.6,
(iii)
—
by
replacing
Π
∗∗
F
by
a
suitable
base-admissible
maximal
almost
pro-Σ
quotient
Π
F
Π
∗∗
F
[i.e.,
a
quotient
as
in
Lemma
2.6,
(iii)]
that
dominates
the
original
Π
F
Π
∗∗
F
and
applying
the
conclusion
“
β
i
∗
(Π
∗
(v
i
)
F
x
)
=
Π
∗
(v
i
)
F
x
”,
together
with
the
property
(a)
discussed
above,
in
the
case
where
“
α
∗
”
is
taken
to
be
α
∗∗∗
∈
Aut(Π
∗∗∗
T
)
—
that
we
may
assume
without
loss
of
generality
that
(‡
5
):
β
i
∗
induces
the
identity
automorphism
of
Π
∗
(v
i
)
F
x
.
∼
Next,
let
Π
∗
e
F
⊆
Π
∗
(v
i
)
F
x
be
a
nodal
subgroup
of
Π
∗G
x
←
Π
∗
F
associated
i
∼
∗
∗
to
e
F
i
∈
Node(G
x
)
that
is
contained
in
Π
∗
(v
i
)
F
x
;
Π
∗
v
new,x
F
;i
⊆
Π
G
x
←
Π
F
a
verticial
subgroup
[which
may
depend
on
i
∈
{1,
2}!]
associated
to
F
v
new,x
∈
Vert(G
x
)
that
contains
Π
∗
e
F
:
i
Π
∗
v
new,x
F
;i
⊇
Π
∗
e
F
i
⊆
Π
∗
(v
i
)
F
x
⊆
∼
Π
∗G
x
←
Π
∗
F
.
42
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Then
it
follows
from
the
inclusion
Π
∗
e
F
⊆
Π
∗
(v
i
)
F
x
,
together
with
(‡
5
),
i
that
β
∗
(Π
∗
F
)
=
Π
∗
F
.
Since,
moreover,
Π
∗
F
is
the
unique
verti-
i
v
new,x
;i
e
i
∼
∗
∗
F
cial
subgroup
of
Π
G
x
←
Π
F
associated
to
v
new,x
that
contains
Π
∗
e
F
[cf.
e
i
i
Proposition
1.7,
(v),
(vi)],
it
follows
immediately
from
the
fact
that
∗
∗
α
F
∗
∈
Out
|grph|
(Π
∗G
x
)
that
β
i
∗
(Π
∗
v
new,x
F
F
;i
)
=
Π
v
new,x
;i
.
Thus,
β
i
preserves
the
closed
subgroup
Π
∗
F
i
⊆
Π
∗
F
of
Π
∗
F
obtained
by
forming
the
image
of
the
natural
homomorphism
∗
∗
lim
Π
∗
v
new,x
←
Π
→
Π
−→
Π
∗
F
F
F
F
(v
i
)
x
;i
e
i
−→
—
where
the
inductive
limit
is
taken
in
the
category
of
profinite
groups.
Next,
let
us
observe
that
the
Π
∗
F
-conjugacy
class
of
Π
∗
F
i
⊆
Π
∗
F
coin-
cides
with
the
Π
∗
F
-conjugacy
class
of
the
image
Π
∗
(G
x
)|
H
[cf.
[CbTpI],
i
Definition
2.2,
(ii)]
of
the
composite
∼
Π
(G
x
)|
H
i
→
Π
G
x
←
Π
F
Π
∗
F
—
where
the
first
arrow
is
the
natural
outer
injection
discussed
in
[CbTpI],
Proposition
2.11.
On
the
other
hand,
if
we
write
Π
∗
(G
x
)
F
for
{e
}
i
the
maximal
almost
pro-Σ
quotient
of
Π
(G
x
)
{e
F
}
[cf.
[CbTpI],
Definition
i
2.8]
determined
by
the
maximal
almost
pro-Σ
quotient
Π
∗G
x
and
the
∼
natural
outer
isomorphism
Φ
(G
x
)
{e
F
}
:
Π
(G
x
)
{e
F
}
→
Π
G
x
[cf.
[CbTpI],
i
i
Definition
2.10],
then
Π
∗
F
i
may
be
regarded
as
a
verticial
subgroup
of
∼
∼
Π
∗
(G
x
)
F
→
Π
∗G
x
←
Π
∗
F
[cf.
[CbTpI],
Proposition
2.9,
(i),
(3)].
Thus,
it
{e
}
i
follows
from
Proposition
1.7,
(vii),
that
Π
∗
F
i
is
commensurably
terminal
in
Π
∗
F
.
Moreover,
by
applying
a
similar
argument
to
the
argument
given
in
[CmbCsp],
Definition
2.1,
(iii),
(vi),
or
[NodNon],
Definition
5.1,
(ix),
(x)
[i.e.,
roughly
speaking,
by
considering
the
portion
of
the
underlying
scheme
X
2
of
X
2
log
corresponding
to
the
underlying
scheme
(X
v
i
)
2
of
log
the
2-nd
log
configuration
space
(X
v
i
)
log
2
of
the
stable
log
curve
X
v
i
determined
by
G|
v
i
—
cf.
[CbTpI],
Definition
2.1,
(iii)],
one
concludes
∼
that
there
exists
a
verticial
subgroup
Π
v
i
⊆
Π
G
←
Π
B
associated
to
v
i
∈
Vert(G)
such
that
the
outer
action
of
Π
v
i
on
Π
∗
F
determined
by
ρ
∗
2/1
the
composite
Π
v
i
→
Π
B
→
Out(Π
∗
F
)
—
where
we
write
ρ
∗
2/1
for
the
outer
action
determined
by
the
exact
sequence
of
profinite
groups
1
−→
Π
∗
F
−→
Π
∗
T
−→
Π
B
−→
1
—
preserves
the
Π
∗
F
-conjugacy
class
of
the
commensurably
terminal
subgroup
Π
∗
F
i
⊆
Π
∗
F
[so
we
obtain
a
natural
outer
representation
Π
v
i
→
Out(Π
∗
F
i
)
—
cf.
[CbTpI],
Lemma
2.12,
(iii)],
and,
moreover,
that
if
we
def
out
write
Π
∗
T
i
=
Π
∗
F
i
Π
v
i
(⊆
Π
∗
T
)
[cf.
the
discussion
entitled
“Topolog-
ical
groups”
in
[CbTpI],
§0],
then
it
follows
from
Proposition
1.7,
(ii),
COMBINATORIAL
ANABELIAN
TOPICS
III
43
that
Π
∗
T
i
is
naturally
isomorphic
to
a
profinite
group
of
the
form
“Π
∗
T
”
obtained
by
taking
“G”
to
be
G|
v
i
.
∗
is
an
automorphism
over
the
quo-
Now
since
β
i
∗
(Π
∗
F
i
)
=
Π
∗
F
i
,
and
α
∼
∗
→
Π
B
,
one
verifies
immediately
that
β
i
determines
an
au-
tient
Π
∗
F
/Π
T
tomorphism
β
T
∗
i
of
Π
∗
T
i
over
Π
v
i
.
Thus,
since
the
quantity
“3g
−
3
+
r”
associated
to
G|
v
i
is
<
3g−3+r,
by
considering
a
diagram
similar
to
the
diagram
of
[CmbCsp],
Definition
2.1,
(vi),
or
[NodNon],
Definition
5.1,
(x),
and
applying
the
induction
hypothesis
[cf.
also
Lemma
2.6,
(ii)],
we
conclude
—
by
replacing
Π
∗∗
F
by
a
suitable
base-admissible
maximal
∗∗
almost
pro-Σ
quotient
Π
F
Π
∗∗
F
that
dominates
the
original
Π
F
Π
F
and
applying
the
conclusion
that
β
i
∗
determines
an
automorphism
of
α
∗
”
is
taken
to
be
α
∗∗∗
∈
Aut(Π
∗∗∗
Π
∗
T
i
over
Π
v
i
in
the
case
where
“
T
)
—
that
we
may
assume
without
loss
of
generality
that
(‡
):
β
∗
is
a
Π
∗
-inner
automorphism.
6
T
i
F
i
In
particular,
it
follows
immediately,
by
allowing
i
∈
{1,
2}
to
vary,
from
Proposition
1.7,
(vii)
[which
implies
the
commensurable
terminal-
ity
of
Π
∗
(v
i
)
F
x
⊆
Π
F
∗
i
],
that
the
outomorphisms
of
Π
∗
(G
x
)
F
,
Π
∗
(G
x
)
F
{e
1
}
obtained
by
conjugating
α
F
∗
by
the
isomorphisms
Π
∗
(G
x
)
[induced
by
Φ
(G
x
)
{e
F
}
],
Π
∗
(G
x
)
1
∼
{e
F
2
}
profinite
Dehn
multi-twists
of
Π
∗
(G
x
)
{e
F
1
}
∼
{e
2
}
→
Π
∗G
x
→
Π
∗G
x
[induced
by
Φ
(G
x
)
{e
F
}
]
are
,
Π
∗
(G
x
)
F
{e
1
}
2
,
respectively.
Thus,
F
{e
2
}
it
follows
from
Lemma
1.10
that
α
F
∗
is
the
identity
outomorphism.
This
completes
the
proof
of
Claim
2.8.C,
hence
also
of
Lemma
2.8.
Theorem
2.9
(Almost
pro-Σ
analogue
of
the
injectivity
por-
tion
of
the
theory
of
combinatorial
cuspidalization).
Let
Σ
be
a
nonempty
set
of
prime
numbers,
n
a
positive
integer,
(g,
r)
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0,
and
X
a
hyperbolic
curve
of
type
(g,
r)
over
an
algebraically
closed
field
of
characteristic
zero.
For
each
positive
integer
i,
write
X
i
for
the
i-th
configuration
space
of
X
[cf.
[MzTa],
Definition
2.1,
(i)];
Π
i
for
the
pro-Primes
configuration
space
group
[cf.
[MzTa],
Definition
2.3,
(i)]
given
by
the
étale
fundamental
group
π
1
(X
i
)
of
X
i
.
Also,
we
shall
write
pr
:
X
n+1
X
n
for
the
projection
obtained
by
forgetting
the
(n
+
1)-
st
factor
and
Π
n+1/n
⊆
Π
n+1
for
the
kernel
of
some
fixed
surjection
pr
Π
:
Π
n+1
Π
n
[that
belongs
to
the
collection
of
surjections
that
con-
stitutes
the
outer
surjection]
induced
by
pr.
Let
Π
n+1
Π
∗
n+1
be
a
quotient
of
Π
n+1
such
that
the
quotient
Π
∗
n+1/n
of
Π
n+1/n
⊆
Π
n+1
de-
termined
by
the
quotient
Π
n+1
Π
∗
n+1
is
a
maximal
almost
pro-Σ
quotient
of
Π
n+1/n
[cf.
Definition
1.1].
Then
there
exists
a
quotient
Π
n+1
Π
∗∗
n+1
of
Π
n+1
such
that
the
following
conditions
are
satisfied:
44
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(i)
The
quotient
Π
n+1
Π
∗∗
n+1
dominates
[cf.
the
discussion
entitled
“Topological
groups”
in
§0]
the
quotient
Π
n+1
Π
∗
n+1
∗
[i.e.,
Π
n+1
Π
∗∗
n+1
Π
n+1
].
(ii)
The
quotient
Π
∗∗
n+1/n
of
Π
n+1/n
⊆
Π
n+1
determined
by
the
quo-
tient
Π
n+1
Π
∗∗
n+1
is
a
maximal
almost
pro-Σ
quotient
of
Π
n+1/n
.
(iii)
Let
α
∗
be
an
outomorphism
of
Π
∗
n+1
and
Π
n+1
Π
∗∗∗
n+1
a
∗∗
quotient
that
dominates
the
quotient
Π
n+1
Π
n+1
and
in-
duces
a
maximal
almost
pro-Σ
quotient
Π
∗∗∗
n+1/n
of
Π
n+1/n
.
∗
Suppose
that
α
arises
from
an
FC-admissible
[cf.
Defi-
∗∗∗
nition
2.1,
(v)]
automorphism
α
∗∗∗
of
Π
∗∗∗
[i.e.,
n+1
over
Π
n
∗∗∗
which
induces
the
identity
automorphism
of
Π
n
]
—
where
we
write
Π
∗∗∗
for
the
quotient
of
Π
n
determined
by
the
quotient
n
∗
Π
n+1
Π
∗∗∗
n+1
.
Then
α
is
the
identity
outomorphism.
Proof.
First,
we
claim
that
the
following
assertion
holds:
Claim
2.9.A:
To
verify
Theorem
2.9,
it
suffices
to
verify
Theorem
2.9
in
the
case
where
the
kernel
of
the
natural
surjection
Π
n+1
Π
∗
n+1
is
contained
in
Π
n+1/n
,
i.e.,
the
natural
surjection
∼
Π
n
←
Π
n+1
/Π
n+1/n
Π
∗
n+1
/Π
∗
n+1/n
—
where
the
first
arrow
is
the
natural
isomorphism
—
is
an
isomorphism.
Indeed,
Claim
2.9.A
follows
immediately,
by
considering
the
objects
obtained
by
base-changing
the
various
objects
that
appear
in
the
case
∼
of
an
arbitrary
quotient
Π
n+1
Π
∗
n+1
via
the
natural
surjection
Π
n
←
Π
n+1
/Π
n+1/n
Π
∗
n+1
/Π
∗
n+1/n
.
By
Claim
2.9.A,
we
may
assume
with-
out
loss
of
generality
that
the
kernel
of
Π
n+1
Π
∗
n+1
is
contained
in
Π
n+1/n
.
Next,
we
claim
that
the
following
assertion
holds:
Claim
2.9.B:
To
verify
Theorem
2.9,
it
suffices
to
verify
Theorem
2.9
in
the
case
where
n
=
1.
Indeed,
suppose
that
n
≥
2,
and
that
Theorem
2.9
holds
whenever
n
=
1.
Write
Π
n+1/n−1
⊆
Π
n+1
for
the
kernel
of
the
outer
surjection
Π
n+1
Π
n−1
induced
by
the
projection
X
n+1
→
X
n−1
obtained
by
forgetting
the
(n
+
1)-st
and
n-th
factors
of
X
n+1
;
Π
n/n−1
⊆
Π
n
for
the
kernel
of
the
outer
surjection
Π
n
Π
n−1
induced
by
the
projection
X
n
→
X
n−1
obtained
by
forgetting
the
n-th
factor
of
X
n
;
Π
∗
n+1/n−1
for
∗
the
quotient
of
Π
n+1/n−1
determined
by
the
quotient
Π
n+1
Π
n+1
.
Then
let
us
recall
[cf.
[MzTa],
Proposition
2.4,
(i)]
that
one
may
in-
terpret
the
surjection
Π
∗
n+1/n−1
Π
n/n−1
induced
by
the
fixed
surjec-
tion
pr
Π
:
Π
n+1
Π
n
as
the
surjection
“pr
Π
:
Π
∗
2
Π
1
”
in
the
case
where
“X”
is
of
type
(g,
r
+
n
−
1).
Thus,
by
applying
Theorem
2.9
COMBINATORIAL
ANABELIAN
TOPICS
III
45
in
the
case
where
n
=
1
to
the
quotient
Π
n+1/n−1
Π
∗
n+1/n−1
,
we
obtain
a
quotient
Π
∗∗
n+1/n−1
of
Π
n+1/n−1
which
satisfies
conditions
(i),
(ii),
(iii)
in
the
statement
of
Theorem
2.9.
[Here,
we
note
that
since
the
kernel
of
Π
n+1/n−1
Π
∗
n+1/n−1
is
contained
in
Π
n+1/n
,
the
kernel
of
Π
n+1/n−1
Π
∗∗
n+1/n−1
is
also
contained
in
Π
n+1/n
.]
Next,
let
N
⊆
Π
n+1/n
be
a
normal
open
subgroup
of
Π
n+1/n
with
respect
to
which
Π
∗∗
n+1/n
is
a
maximal
almost
pro-Σ
quotient
of
Π
n+1/n
.
Then
it
follows
immediately
from
Lemma
1.2,
(iii)
[cf.
also
[MzTa],
Proposition
2.2,
(ii)],
that
we
may
assume
without
loss
of
generality
—
by
replacing
N
by
a
suitable
normal
open
subgroup
contained
in
N
—
∗∗
that
the
kernel
of
Π
n+1/n
Π
∗∗
n+1/n
is
normal
in
Π
n+1
.
Write
Π
n+1
for
the
quotient
of
Π
n+1
by
the
kernel
of
Π
n+1/n
Π
∗∗
n+1/n
.
Then
it
is
im-
∗∗
mediate
that
this
quotient
Π
n+1
satisfies
conditions
(i),
(ii)
in
the
state-
ment
of
Theorem
2.9,
and,
moreover,
that
the
kernel
of
Π
n+1
Π
∗∗
n+1
is
satisfies
condition
(iii)
in
the
contained
in
Π
n+1/n
.
To
verify
that
Π
∗∗
n+1
∗∗∗
statement
of
Theorem
2.9,
let
Π
n+1
Π
n+1
be
a
quotient
as
in
condi-
tion
(iii)
in
the
statement
of
Theorem
2.9
and
α
∗
an
automorphism
of
∗∗∗
of
Π
∗∗∗
Π
∗
n+1
which
arises
from
an
FC-admissible
automorphism
α
n+1
∗∗∗
over
Π
n
.
Then
since
α
is
FC-admissible,
it
is
immediate
that
α
∗∗∗
∗∗∗
∗∗∗
preserves
Π
∗∗∗
n+1/n−1
⊆
Π
n+1
,
where
we
write
Π
n+1/n−1
for
the
quotient
of
Π
n+1/n−1
determined
by
the
quotient
Π
n+1
Π
∗∗∗
n+1
.
In
particular,
it
,
together
with
the
fact
that
α
∗∗∗
is
follows
from
our
choice
of
Π
∗∗
n+1/n−1
∗
is
an
automor-
an
automorphism
of
Π
∗∗∗
n+1
over
Π
n
[which
implies
that
α
∗
phism
of
Π
n+1
over
Π
n
],
that
we
may
assume
without
loss
of
generality
—
i.e.,
by
replacing
α
∗
by
a
suitable
Π
∗
n+1/n−1
-conjugate,
which
may
in
fact
[in
light
of
the
slimness
of
Π
n/n−1
—
cf.,
e.g.,
[CmbGC],
Remark
1.1.3]
be
taken
to
be
a
Π
∗
n+1/n
-conjugate
—
that
the
automorphism
of
∗
is
the
identity
automorphism.
Thus,
since
α
∗
is
an
Π
∗
n+1/n
induced
by
α
automorphism
of
Π
∗
n+1
over
Π
n
,
and
Π
∗
n+1/n
is
slim
[cf.
Proposition
1.7,
∼
out
(i)],
we
may
apply
the
natural
isomorphism
Π
∗
n+1
→
Π
∗
n+1/n
Π
n
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0]
to
conclude
[cf.,
e.g.,
[Hsh],
Lemma
4.10]
that
the
automorphism
α
∗
of
Π
∗
n+1
is
the
identity
automorphism.
In
particular,
we
conclude
that
Π
∗∗
n+1
satisfies
condition
(iii)
in
the
statement
of
Theorem
2.9.
This
completes
the
proof
of
Claim
2.9.B.
By
Claim
2.9.B,
we
may
assume
without
loss
of
generality
that
n
=
1.
On
the
other
hand,
if
n
=
1,
then
one
verifies
easily
that
Theorem
2.9
follows
immediately
from
Lemma
2.8.
This
completes
the
proof
of
Theorem
2.9.
Corollary
2.10
(Almost
pro-l
analogue
of
the
injectivity
por-
tion
of
the
theory
of
combinatorial
cuspidalization).
Let
l
be
a
46
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
prime
number,
n
a
positive
integer,
(g,
r)
a
pair
of
nonnegative
in-
tegers
such
that
2g
−
2
+
r
>
0,
and
X
a
hyperbolic
curve
of
type
(g,
r)
over
an
algebraically
closed
field
of
characteristic
zero.
For
each
positive
integer
i,
write
X
i
for
the
i-th
configuration
space
of
X
[cf.
[MzTa],
Definition
2.1,
(i)];
Π
i
for
the
pro-Primes
configu-
ration
space
group
[cf.
[MzTa],
Definition
2.3,
(i)]
given
by
the
étale
fundamental
group
π
1
(X
i
)
of
X
i
.
Let
Π
n+1
Π
∗
n+1
be
an
F-
characteristic
SA-maximal
almost
pro-l
quotient
of
Π
n+1
[cf.
Definition
2.1,
(ii),
(iii)].
Then
there
exists
an
F-characteristic
SA-
maximal
almost
pro-l
quotient
Π
n+1
Π
∗∗
n+1
of
Π
n+1
such
that
Π
n+1
Π
∗∗
dominates
[cf.
the
discussion
entitled
“Topological
n+1
∗
groups”
in
§0]
the
quotient
Π
n+1
Π
n+1
,
and,
moreover,
satisfies
the
following
property:
For
any
F-characteristic
SA-maximal
almost
pro-l
quotient
Π
n+1
Π
∗∗∗
n+1
of
Π
n+1
that
dominates
the
quotient
,
the
image
of
the
composite
Π
n+1
Π
∗∗
n+1
FC
FC
∗
∗∗∗
∗∗∗
Π
)
∩
Ker
Out
(Π
)
→
Out
(Π
)
Out
FC
(Π
∗∗∗
n+1
n+1
n+1
n
FC
FC
∗
∗
∗∗∗
∗
→
Out
FC
(Π
∗∗∗
n+1
Π
n+1
)
Out
(Π
n+1
Π
n+1
)
→
Out
(Π
n+1
)
[cf.
Definition
2.1,
(vii),
(viii)]
—
where
we
write
Π
∗∗∗
n
for
the
quotient
,
and
the
homomor-
of
Π
n
determined
by
the
quotient
Π
n+1
Π
∗∗∗
n+1
FC
FC
∗∗∗
∗∗∗
phism
Out
(Π
n+1
)
→
Out
(Π
n
)
[in
large
parentheses]
is
the
homo-
morphism
induced
by
the
projection
X
n+1
→
X
n
obtained
by
forgetting
the
(n
+
1)-st
factor
—
is
trivial.
Proof.
This
follows
immediately
from
Theorem
2.9,
together
with
Propo-
sition
2.3,
(ii).
Remark
2.10.1.
(i)
Theorem
2.9
and
Corollary
2.10
may
be
regarded,
respectively,
as
almost
pro-Σ,
almost
pro-l
versions
of
the
injectivity
portion
of
[NodNon],
Theorem
B.
In
this
context,
it
is
of
interest
to
recall
that
the
pro-l
version
of
this
sort
of
injectivity
result
may
also
be
obtained
by
means
of
the
Lie-theoretic
approach
of
[Tk].
On
the
other
hand,
it
does
not
appear,
at
the
time
of
writing,
that
this
Lie-theoretic
approach
may
be
extended
so
as
to
yield
an
alternate
proof
either
of
the
profinite
portion
of
the
injectivity
result
of
[NodNon],
Theorem
B,
or
of
the
almost
pro-Σ/pro-l
versions
of
this
result
given
in
Theorem
2.9,
Corollary
2.10
of
the
present
paper.
(ii)
In
the
context
of
the
observations
of
(i),
it
is
of
interest
to
recall
that
the
various
injectivity
results
of
[NodNon]
and
the
present
paper
that
are
discussed
in
(i)
are
obtained
as
consequences
of
COMBINATORIAL
ANABELIAN
TOPICS
III
47
various
combinatorial
versions
of
the
Grothendieck
Conjecture.
From
this
point
of
view,
it
seems
natural
to
pose
the
following
question:
Is
it
possible
to
prove
a
Lie-theoretic
combinatorial
version
of
the
Grothendieck
Conjecture
that
allows
one
to
derive
the
Lie-theoretic
injectivity
results
of
[Tk]
by
means
of
techniques
analogous
to
the
tech-
niques
applied
in
[NodNon]
and
the
present
paper?
At
the
time
of
writing,
it
is
not
clear
to
the
authors
whether
or
not
this
question
may
be
answered
in
the
affirmative.
In
the
remainder
of
§2,
we
consider
an
almost
pro-l
analogue
of
the
tripod
homomorphism
of
[CbTpII],
Definition
3.19.
Here,
we
recall
that,
as
discussed
in
[CbTpII],
Remark
3.19.1,
the
tripod
homomorphism
may
be
understood
as
a
sort
of
abstract
combinatorial
analogue
of
the
natural
surjection
from
the
arithmetric
fundamental
group
of
a
moduli
stack
of
curves
over
an
arithmetic
base
field
to
the
absolute
Galois
group
of
the
base
field.
Lemma
2.11
(Commensurators
of
various
subgroups
of
geo-
metric
origin).
We
shall
apply
the
notational
conventions
established
in
§3
of
[CbTpII].
In
the
notation
of
[CbTpII],
Lemma
3.6,
sup-
pose
that
(j,
i)
=
(1,
2);
E
=
{i,
j};
z
i,j,x
∈
Edge(G
j∈E\{i},x
).
[Thus,
∼
G
j∈E\{i},x
=
G
i∈E\{j},x
=
G;
Π
2
=
Π
E
;
Π
1
=
Π
{j}
→
Π
G
j∈E\{i},x
=
Π
G
;
∼
def
def
Π
2/1
=
Π
E/(E\{i})
→
Π
G
i∈E,x
.]
Write
G
2/1
=
G
i∈E,x
;
G
1\2
=
G
j∈E,x
;
def
def
∼
Π
Π
p
Π
1\2
=
p
E/{2}
:
Π
2
Π
{2}
;
Π
1\2
=
Ker(p
1\2
)
=
Π
E/{2}
→
Π
G
1\2
;
def
def
z
x
=
z
i,j,x
∈
Edge(G);
c
diag
=
c
diag
i,j,x
∈
Cusp(G
2/1
)
[cf.
the
notation
def
new
∈
Vert(G
2/1
)
[cf.
the
no-
of
[CbTpII],
Lemma
3.6,
(ii)];
v
new
=
v
i,j,x
tation
of
[CbTpII],
Lemma
3.6,
(iv)].
Let
Π
z
x
⊆
Π
1
be
an
edge-like
subgroup
associated
to
z
x
∈
Edge(G);
Π
v
new
⊆
Π
2/1
a
verticial
sub-
group
associated
to
v
new
;
Π
c
diag
⊆
Π
2/1
a
cuspidal
subgroup
associated
to
c
diag
that
is
contained
in
Π
v
new
[cf.
[CbTpII],
Lemma
3.6,
(iv)].
Let
Π
2
Π
∗
2
be
an
SA-maximal
almost
pro-l
quotient
of
Π
2
[cf.
Definition
2.1,
(ii)].
Write
Π
∗
2/1
,
Π
∗
1\2
,
Π
∗
1
,
Π
∗{2}
for
the
respective
quo-
tients
of
Π
2/1
,
Π
1\2
,
Π
1
,
Π
{2}
determined
by
the
quotient
Π
2
Π
∗
2
of
Π
2
;
Π
∗G
,
Π
∗G
2/1
for
the
respective
quotients
of
Π
G
,
Π
G
2/1
determined
by
∼
the
quotients
Π
1
Π
∗
1
,
Π
2/1
Π
∗
2/1
and
the
isomorphisms
Π
1
→
Π
G
,
∼
∗
∗
∗
Π
2/1
→
Π
G
2/1
fixed
in
[CbTpII],
Definition
3.1,
(iii);
(p
Π
2/1
)
:
Π
2
Π
1
,
∗
∗
∗
(p
Π
1\2
)
:
Π
2
Π
{2}
for
the
respective
natural
surjections
induced
by
Π
∗
∗
∗
∗
∗
p
Π
2/1
:
Π
2
Π
1
,
p
1\2
:
Π
2
Π
{2}
;
Π
z
x
⊆
Π
1
,
Π
c
diag
⊆
Π
v
new
⊆
Π
2/1
for
the
respective
images
of
Π
z
x
⊆
Π
1
,
Π
c
diag
⊆
Π
v
new
⊆
Π
2/1
in
Π
∗
1
,
48
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
def
def
def
Π
∗
2/1
;
Π
∗
2
|
z
x
=
Π
∗
2
×
Π
∗
1
Π
∗
z
x
⊆
Π
∗
2
;
D
c
∗
diag
=
N
Π
∗
2
(Π
∗
c
diag
);
I
v
∗
new
|
z
x
=
def
Z
Π
∗
2
|
zx
(Π
∗
v
new
)
⊆
D
v
∗
new
|
z
x
=
N
Π
∗
2
|
zx
(Π
∗
v
new
).
Then
the
following
hold:
(i)
It
holds
that
D
c
∗
diag
∩
Π
∗
2/1
=
C
Π
∗
2
(Π
∗
c
diag
)
∩
Π
∗
2/1
=
Π
∗
c
diag
.
(ii)
It
holds
that
C
Π
∗
2
(Π
∗
c
diag
)
=
D
c
∗
diag
.
∗
∗
∗
(iii)
The
surjection
(p
Π
2/1
)
:
Π
2
Π
1
determines
an
isomorphism
∼
D
c
∗
diag
/Π
∗
c
diag
→
Π
∗
1
.
Moreover,
the
composite
∼
Π
1
Π
1
∗
←
D
c
∗
diag
/Π
∗
c
diag
Π
∗{2}
—
where
the
first
arrow
is
the
natural
surjection,
the
second
arrow
is
the
isomorphism
obtained
above,
and
the
third
arrow
∗
∗
∗
is
the
surjection
determined
by
(p
Π
1\2
)
:
Π
2
Π
{2}
—
coin-
cides,
up
to
composition
with
an
inner
automorphism,
with
the
natural
surjection
Π
1
Π
∗{2}
.
(iv)
The
composite
I
v
∗
new
|
z
x
→
D
v
∗
new
|
z
x
→
Π
∗
z
x
is
an
isomorphism.
(v)
The
natural
inclusions
Π
∗
v
new
,
I
v
∗
new
|
z
x
→
D
v
∗
new
|
z
x
determine
an
∼
isomorphism
Π
∗
v
new
×
I
v
∗
new
|
z
x
→
D
v
∗
new
|
z
x
=
C
Π
∗
2
|
zx
(Π
∗
v
new
).
(vi)
It
holds
that
C
Π
∗
2
(D
v
∗
new
|
z
x
)
⊆
C
Π
∗
2
(Π
∗
v
new
).
(vii)
D
v
∗
new
|
z
x
is
commensurably
terminal
in
Π
∗
2
.
Proof.
First,
we
verify
assertion
(i).
Observe
that
we
have
inclusions
Π
∗
c
diag
⊆
D
c
∗
diag
⊆
C
Π
∗
2
(Π
∗
c
diag
).
Thus,
since
Π
c
∗
diag
is
commensurably
ter-
minal
in
Π
∗
2/1
[cf.
Proposition
1.7,
(vii)],
we
conclude
that
Π
∗
c
diag
⊆
D
c
∗
diag
∩
Π
∗
2/1
⊆
C
Π
∗
2
(Π
∗
c
diag
)
∩
Π
∗
2/1
=
C
Π
∗
2/1
(Π
∗
c
diag
)
=
Π
∗
c
diag
.
This
com-
pletes
the
proof
of
assertion
(i).
Assertions
(ii),
(iii)
follow
immediately
from
assertion
(i),
together
with
the
[easily
verified]
fact
that
the
com-
)
∗
(p
Π
2/1
posite
D
c
∗
diag
→
Π
∗
2
Π
∗
1
is
surjective.
Next,
we
verify
assertion
(iv).
Since
Π
∗
v
new
is
slim
and
commensurably
terminal
in
Π
∗
2/1
[cf.
Proposition
1.7,
(ii),
(vii)],
it
follows
that
I
v
∗
new
|
z
x
∩
Π
∗
2/1
=
{1},
which
implies
the
injectivity
of
the
composite
in
question.
On
the
other
hand,
since
the
composite
I
v
new
|
z
x
→
D
v
new
|
z
x
→
Π
2
|
z
x
Π
z
x
is
surjective
[cf.
[CbTpII],
Lemma
3.11,
(iv)],
it
follows
immediately
that
the
composite
I
v
∗
new
|
z
x
→
D
v
∗
new
|
z
x
→
Π
∗
2
|
z
x
Π
∗
z
x
is
surjective.
This
completes
the
proof
of
assertion
(iv).
Next,
we
verify
assertion
(v).
It
follows
immediately
from
asser-
tion
(iv),
together
with
the
commensurable
terminality
of
Π
∗
v
new
in
Π
∗
2/1
[cf.
Proposition
1.7,
(vii)],
that
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Π
∗
v
new
−→
D
v
∗
new
|
z
x
−→
Π
∗
z
x
−→
1
COMBINATORIAL
ANABELIAN
TOPICS
III
49
—
where
we
observe
that
the
inclusion
I
v
∗
new
|
z
x
→
D
v
∗
new
|
z
x
determines
a
splitting
of
this
exact
sequence.
Thus,
it
follows
from
the
definition
of
I
v
∗
new
|
z
x
that
the
natural
inclusions
Π
∗
v
new
,
I
v
∗
new
|
z
x
→
D
v
∗
new
|
z
x
deter-
∼
mine
an
isomorphism
Π
∗
v
new
×
I
v
∗
new
|
z
x
→
D
v
∗
new
|
z
x
.
On
the
other
hand,
again
by
the
commensurable
terminality
of
Π
∗
v
new
in
Π
∗
2/1
[cf.
Proposi-
tion
1.7,
(vii)],
the
above
displayed
sequence
implies
that
D
v
∗
new
|
z
x
=
C
Π
∗
2
|
zx
(Π
∗
v
new
).
This
completes
the
proof
of
assertion
(v).
Next,
we
verify
assertion
(vi).
It
follows
from
the
commensurable
terminality
of
Π
∗
v
new
in
Π
∗
2/1
[cf.
Proposition
1.7,
(vii)]
that
D
v
∗
new
|
z
x
∩
Π
∗
2/1
=
Π
∗
v
new
.
Thus,
since
Π
∗
2/1
is
normal
in
Π
∗
2
,
assertion
(vi)
follows
immediately
from
[CbTpII],
Lemma
3.9,
(i).
This
completes
the
proof
of
assertion
(vi).
Finally,
we
verify
assertion
(vii).
Since
Π
∗
z
x
⊆
Π
∗
1
is
commensu-
rably
terminal
in
Π
∗
1
[cf.
Proposition
1.7,
(vii)],
it
follows
from
the
surjectivity
of
the
composite
D
v
∗
new
|
z
x
→
Π
∗
2
|
z
x
Π
∗
z
x
[cf.
asser-
tion
(iv)]
that
C
Π
∗
2
(D
v
∗
new
|
z
x
)
⊆
Π
∗
2
|
z
x
.
In
particular,
it
follows
im-
mediately
from
assertions
(v),
(vi)
that
D
v
∗
new
|
z
x
⊆
C
Π
∗
2
(D
v
∗
new
|
z
x
)
⊆
C
Π
∗
2
(Π
∗
v
new
)
∩
Π
∗
2
|
z
x
=
C
Π
∗
2
|
zx
(Π
∗
v
new
)
=
D
v
∗
new
|
z
x
.
This
completes
the
proof
of
assertion
(vii).
Lemma
2.12
(Commensurator
of
a
tripod
arising
from
an
edge).
In
the
notation
of
Lemma
2.11,
let
Π
2
Π
∗∗
2
be
an
SA-
maximal
almost
pro-l
quotient
of
Π
2
[cf.
Definition
2.1,
(ii)]
that
dominates
Π
2
Π
∗
2
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
We
shall
use
similar
notation
∗∗
∗∗
∗∗
∗∗
∗∗
Π
∗∗
2/1
;
Π
1\2
;
Π
1
;
Π
{2}
;
Π
G
;
Π
G
2/1
;
∗∗
∗∗
∗∗
Π
∗∗
∗∗
∗∗
(p
Π
2/1
)
:
Π
2
Π
1
;
(p
1\2
)
:
Π
2
Π
{2}
;
∗∗
∗∗
∗∗
∗∗
Π
∗∗
z
x
⊆
Π
1
;
Π
c
diag
⊆
Π
v
new
⊆
Π
2/1
;
∗∗
∗∗
∗∗
Π
∗∗
2
|
z
x
;
D
c
diag
;
I
v
new
|
z
x
⊆
D
v
new
|
z
x
for
objects
associated
to
Π
2
Π
∗∗
2
to
the
notation
introduced
in
the
statement
of
Lemma
2.11
for
objects
associated
to
Π
2
Π
∗
2
.
Suppose
that
the
natural
[outer]
surjection
Π
1
Π
∗∗
{2}
dominates
the
quotient
Π
1
Π
∗
1
.
Then
the
following
hold:
∗
(i)
The
natural
surjection
Π
∗∗
2
Π
2
determines
a
surjection
∗∗
∗
I
v
new
|
z
x
I
v
new
|
z
x
.
∗∗
(ii)
The
image
of
Z
Π
loc
∗∗
(Π
∗∗
v
new
)
⊆
Π
2
[cf.
the
discussion
entitled
2
“Topological
groups”
in
[CbTpII],
§0]
in
Π
∗
2
coincides
with
I
v
∗
new
|
z
x
.
∗∗
∗
∗
(iii)
The
image
of
C
Π
∗∗
(Π
∗∗
v
new
)
⊆
Π
2
in
Π
2
is
contained
in
D
v
new
|
z
x
.
2
50
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∗∗
(iv)
The
natural
outer
action,
by
conjugation,
of
N
Π
∗∗
(Π
∗∗
v
new
)
⊆
Π
2
2
∗
on
[not
Π
∗∗
v
new
but]
Π
v
new
is
trivial.
Proof.
First,
we
verify
assertion
(i).
Observe
that
it
is
immediate
that
∗
∗
the
image
of
I
v
∗∗
new
|
z
x
⊆
Π
∗∗
2
in
Π
2
is
contained
in
I
v
new
|
z
x
.
Thus,
asser-
tion
(i)
follows
immediately
from
Lemma
2.11,
(iv),
together
with
the
∗
[easily
verified]
fact
that
the
natural
surjection
Π
∗∗
2
Π
2
determines
a
∗
surjection
Π
∗∗
z
x
Π
z
x
.
This
completes
the
proof
of
assertion
(i).
∗
(Π
∗∗
Next,
we
verify
assertion
(ii).
Write
Im(Z
Π
loc
∗∗
v
new
))
⊆
Π
2
for
the
2
∗∗
∗
(Π
∗∗
image
of
Z
Π
loc
∗∗
v
new
)
⊆
Π
2
in
Π
2
.
Then
it
follows
from
assertion
(i)
2
(Π
∗∗
that
I
v
∗
new
|
z
x
⊆
Im(Z
Π
loc
∗∗
v
new
)).
Thus,
to
complete
the
verification
of
2
∗
(Π
∗∗
assertion
(ii),
it
suffices
to
verify
that
Im(Z
Π
loc
∗∗
v
new
))
⊆
I
v
new
|
z
x
.
To
2
this
end,
let
us
observe
that
it
follows
immediately
from
the
final
por-
∗∗
∗∗
tion
of
[CbTpII],
Lemma
3.6,
(iv),
that
the
image
(p
Π
1\2
)
(Π
v
new
)
⊆
∼
Π
∗∗
{2}
coincides
with
the
image
of
an
edge-like
subgroup
of
Π
G
←
Π
1
associated
to
z
x
∈
Edge(G)
via
the
natural
[outer]
surjection
Π
1
Π
∗∗
{2}
,
hence
that
the
image
[which
is
well-defined
up
to
conjugacy]
of
∗∗
∗∗
∗∗
∗
(p
Π
1\2
)
(Π
v
new
)
⊆
Π
{2}
in
Π
1
[where
we
recall
that
we
have
assumed
∗
that
Π
1
Π
∗∗
{2}
dominates
Π
1
Π
1
]
is
an
edge-like
subgroup
of
∼
Π
∗G
←
Π
∗
1
associated
to
z
x
∈
Edge(G).
Thus,
since
every
edge-like
subgroup
of
Π
∗
1
is
commensurably
terminal
[cf.
Proposition
1.7,
(vii)],
it
follows
that
the
image
[which
is
well-defined
up
to
conjugacy]
of
∗∗
loc
∗∗
∗
(p
Π
(Π
∗∗
v
new
))
⊆
Π
{2}
in
Π
1
is
contained
in
an
edge-like
subgroup
1\2
)
(Z
Π
∗∗
2
∼
of
Π
∗G
←
Π
∗
1
associated
to
z
x
∈
Edge(G).
On
the
other
hand,
since
loc
loc
Π
∗∗
⊆
Π
∗∗
(Π
∗∗
(Π
∗∗
)
⊆
C
Π
∗∗
(Π
∗∗
)
=
v
new
,
we
have
Z
Π
∗∗
v
new
)
⊆
Z
Π
∗∗
c
diag
c
diag
c
diag
2
2
2
D
c
∗∗
diag
[cf.
Lemma
2.11,
(ii)].
In
particular,
it
follows
immediately
∗∗
loc
∗∗
from
Lemma
2.11,
(iii),
that
the
image
of
(p
Π
(Π
∗∗
v
new
))
⊆
Π
1
2/1
)
(Z
Π
∗∗
2
in
Π
∗
1
is
contained
in
some
Π
∗
1
-conjugate
of
Π
∗
z
x
⊆
Π
∗
1
,
hence
[since
∗∗
(Π
∗∗
I
v
∗∗
new
|
z
x
⊆
Z
Π
loc
∗∗
v
new
)
surjects
onto
Π
z
x
—
cf.
Lemma
2.11,
(iv)]
that
2
∗∗
loc
∗∗
∗
∗
∗
(Π
∗∗
the
image
of
(p
Π
v
new
))
⊆
Π
1
in
Π
1
coincides
with
Π
z
x
⊆
Π
1
2/1
)
(Z
Π
∗∗
2
∗
[cf.
Proposition
1.7,
(v)],
i.e.,
Im(Z
Π
loc
∗∗
(Π
∗∗
v
new
))
⊆
Π
2
|
z
x
.
Thus,
since
[as
2
loc
∗
(Π
∗∗
is
easily
verified]
Im(Z
Π
loc
∗∗
v
new
))
⊆
Z
Π
∗
2
(Π
v
new
),
we
conclude
that
2
∗
loc
∗
loc
∗
∗
Im(Z
Π
loc
∗∗
(Π
∗∗
v
new
))
⊆
Π
2
|
z
x
∩
Z
Π
∗
2
(Π
v
new
)
=
Z
Π
∗
2
|
zx
(Π
v
new
)
=
I
v
new
|
z
x
2
[where
the
final
equality
follows
from
Lemma
2.11,
(v),
together
with
the
slimness
portion
of
Proposition
1.7,
(ii)].
This
completes
the
proof
of
assertion
(ii).
∗
Next,
we
verify
assertion
(iii).
Write
Im(C
Π
∗∗
(Π
∗∗
v
new
))
⊆
Π
2
for
the
2
∗∗
∗
(Π
∗∗
image
of
C
Π
∗∗
v
new
)
⊆
Π
2
in
Π
2
.
Then
it
follows
from
[CbTpII],
2
Lemma
3.9,
(ii),
that
C
Π
∗∗
(Π
∗∗
(Z
Π
loc
∗∗
(Π
∗∗
v
new
)
⊆
N
Π
∗∗
v
new
));
thus,
it
fol-
2
2
2
∗∗
∗∗
lows
from
assertion
(ii)
that
Im(C
Π
2
(Π
v
new
))
⊆
N
Π
2
∗
(I
v
∗
new
|
z
x
).
In
par-
ticular,
since
D
v
∗
new
|
z
x
is
topologically
generated
by
Π
∗
v
new
,
I
v
∗
new
|
z
x
[cf.
COMBINATORIAL
ANABELIAN
TOPICS
III
51
Lemma
2.11,
(v)],
we
conclude
that
∗
∗
(Π
∗∗
Im(C
Π
∗∗
v
new
))
⊆
C
Π
∗
2
(D
v
new
|
z
x
)
=
D
v
new
|
z
x
2
[cf.
Lemma
2.11,
(vii)].
This
completes
the
proof
of
assertion
(iii).
Assertion
(iv)
follows
immediately
from
assertion
(iii),
together
with
Lemma
2.11,
(v).
This
completes
the
proof
of
Lemma
2.12.
Corollary
2.13
(Almost
pro-l
quotients
and
tripod
homomor-
phisms).
In
the
notation
of
Definition
2.1,
suppose
that
n
≥
3.
Let
Π
tpd
⊆
Π
3
be
a
1-central
[{1,
2,
3}-]tripod
of
Π
3
[cf.
[CbTpII],
Defi-
nitions
3.3,
(i);
3.7,
(ii)];
Π
tpd
(Π
tpd
)
‡
an
almost
pro-l
quotient.
Then
the
following
hold:
(i)
There
exists
an
F-characteristic
SA-maximal
almost
pro-
l
quotient
[cf.
Definition
2.1,
(ii),
(iii)]
Π
∗
n
of
Π
n
that
satisfies
the
following
condition:
If
we
write
Π
∗
3
for
the
quotient
of
Π
3
determined
by
the
quotient
Π
n
Π
∗
n
and
(Π
tpd
)
∗
⊆
Π
∗
3
for
the
image
of
Π
tpd
⊆
Π
3
in
Π
∗
3
,
then
the
quotient
Π
tpd
(Π
tpd
)
∗
dominates
the
quotient
Π
tpd
(Π
tpd
)
‡
[cf.
the
discussion
entitled
“Topological
groups”
in
§0].
(ii)
Every
element
of
the
image
Im(T
Π
tpd
)
⊆
Out(Π
tpd
)
of
the
tri-
pod
homomorphism
T
Π
tpd
:
Out
FC
(Π
n
)
−→
Out
C
(Π
tpd
)
associated
to
Π
n
[cf.
[CbTpII],
Definition
3.19]
preserves
the
kernel
of
the
surjection
Π
tpd
(Π
tpd
)
∗
of
(i).
Thus,
we
obtain
a
natural
homomorphism
Im(T
Π
tpd
)
−→
Out((Π
tpd
)
∗
)
.
(iii)
There
exists
an
F-characteristic
SA-maximal
almost
pro-
∗
l
quotient
Π
n
Π
∗∗
n
of
Π
n
that
dominates
Π
n
Π
n
[cf.
(i)]
such
that
the
composite
Out
FC
(Π
n
)
Im(T
Π
tpd
)
→
Out((Π
tpd
)
∗
)
—
where
the
first
arrow
is
the
homomorphism
induced
by
T
Π
tpd
;
the
second
arrow
is
the
homomorphism
of
(ii)
—
factors
through
the
natural
surjection
Out
FC
(Π
n
)
Out
FC
(Π
∗∗
n
Π
n
)
[cf.
Definition
2.1,
(viii);
Remark
2.1.1].
Thus,
we
have
a
natural
commutative
diagram
of
profinite
groups
Out
FC
(Π
n
)
⏐
⏐
−−−→
Im(T
Π
tpd
)
⏐
⏐
tpd
∗
)
)
.
Out
FC
(Π
∗∗
n
Π
n
)
−−−→
Out((Π
52
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Proof.
Assertion
(i)
is
a
consequence
of
Proposition
2.3,
(iii).
Asser-
tion
(ii)
follows
immediately
from
the
fact
that
Π
∗
n
is
F-characteristic,
together
with
the
definition
of
T
Π
tpd
.
Finally,
we
verify
assertion
(iii).
Let
us
first
observe
that
it
follows
immediately
from
the
definition
of
T
Π
tpd
,
together
with
Proposition
2.3,
(ii)
[where
we
observe
that
any
closed
subgroup
of
a
finite
product
of
almost
pro-l
groups
is
almost
pro-l],
that,
to
verify
assertion
(iii),
it
suffices
to
verify
the
following
assertion:
Claim
2.13.A:
There
exists
an
F-characteristic
SA-
maximal
almost
pro-l
quotient
Π
3
Π
∗∗
3
of
Π
3
that
dominates
Π
3
Π
∗
3
such
that
if
we
write
(Π
tpd
)
∗∗
⊆
tpd
Π
∗∗
⊆
Π
3
in
Π
∗∗
3
for
the
image
of
Π
3
,
then
any
auto-
tpd
∗
morphism
of
(Π
)
determined
by
conjugating
by
an
element
γ
∗∗
∈
N
Π
∗∗
((Π
tpd
)
∗∗
)
3
is
(Π
tpd
)
∗
-inner.
To
verify
Claim
2.13.A,
let
Π
3
Π
∗∗
3
be
an
F-characteristic
SA-
maximal
almost
pro-l
quotient
of
Π
3
that
dominates
Π
3
Π
∗
3
and
((Π
tpd
)
∗∗
).
Then
it
follows
immediately
from
[CmbCsp],
γ
∗∗
∈
N
Π
∗∗
3
Proposition
1.9,
(i),
that
Z
Π
3
(Π
tpd
)
⊆
Π
3
surjects
onto
Π
1
,
hence
also
∗∗
onto
Π
∗∗
1
—
where
we
write
Π
1
for
the
quotient
of
Π
1
determined
∗∗
by
the
quotient
Π
3
Π
3
.
In
particular,
there
exists
an
element
τ
∈
Z
Π
3
(Π
tpd
)
such
that
the
images
of
γ
∗∗
and
τ
in
Π
∗∗
1
coincide.
Thus,
by
replacing
γ
∗∗
by
the
difference
of
γ
∗∗
and
the
image
of
τ
in
Π
∗∗
3
,
∗∗
∗∗
we
may
assume
without
loss
of
generality
that
γ
∈
Π
3/1
—
where
we
write
Π
∗∗
3/1
for
the
quotient
of
Π
3/1
[cf.
Definition
2.1]
induced
by
the
quotient
Π
3
Π
∗∗
3
.
In
particular,
the
existence
of
an
F-characteristic
SA-maximal
almost
pro-l
quotient
Π
3
Π
∗∗
3
as
in
Claim
2.13.A
follows
immediately,
in
light
of
Proposition
2.3,
(ii),
from
Lemma
2.12,
(iv).
This
completes
the
proof
of
assertion
(ii).
Finally,
before
proceeding,
we
review
the
following
well-known
result.
Lemma
2.14
(Automorphisms
of
stable
log
curves).
Let
l
be
a
def
def
prime
number.
Write
l
aut
=
l
if
l
is
odd;
l
aut
=
4
if
l
is
even.
If
G
is
a
profinite
group,
then
we
shall
refer
to
the
tensor
product
with
Z/l
aut
Z
of
the
abelianization
of
G
as
the
l
aut
-abelianization
of
G.
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0.
(i)
Let
k
be
an
algebraically
closed
field
such
that
l
is
invertible
in
k,
(Spec
k)
log
the
log
scheme
obtained
by
equipping
Spec
k
with
the
log
structure
determined
by
the
fs
chart
N
→
k
that
maps
1
→
0,
X
log
a
stable
log
curve
over
(Spec
k)
log
,
and
α
an
automorphism
of
X
log
over
(Spec
k)
log
.
Write
Π
1
for
COMBINATORIAL
ANABELIAN
TOPICS
III
53
the
maximal
pro-l
quotient
of
the
kernel
of
the
natural
outer
surjection
π
1
(X
log
)
π
1
((Spec
k)
log
).
Suppose
that
α
acts
trivially
on
the
l
aut
-abelianization
of
Π
1
.
Then
α
is
the
identity
automorphism.
(ii)
Write
M
log
for
the
moduli
stack
of
pointed
stable
curves
of
type
(g,
r)
over
Z[1/l],
where
we
regard
the
marked
points
as
unordered,
equipped
with
the
log
structure
determined
by
the
divisor
at
infinity,
and
C
log
→
M
log
for
the
tautological
stable
log
curve
over
M
log
.
Write
N
log
→
M
log
for
the
finite
log
étale
morphism
of
log
regular
log
stacks
determined
by
the
local
system
of
trivializations
of
the
l
aut
-abelianizations
of
the
log
fundamental
groups
of
the
various
logarithmic
fibers
of
C
log
→
M
log
.
Then
the
underlying
algebraic
stack
N
of
N
log
is
an
algebraic
space.
Proof.
First,
we
consider
assertion
(i).
We
begin
by
recalling
that
when
X
log
is
a
smooth
log
curve,
and
r
≤
1
[so
g
≥
1],
assertion
(i)
follows
immediately
from
classical
theory
of
endomorphisms
of
semi-abelian
va-
rieties
and
automorphisms
of
stable
curves
[cf.,
e.g.,
[Des],
Lemme
5.17;
[DM],
Theorems
1.11,
1.13],
together
with
[in
the
case
where
l
=
2]
the
well-known
fact
that
every
root
of
unity
ζ
such
that
(ζ
−
1)/l
aut
is
an
algebraic
integer
is
necessarily
equal
to
1.
Now
let
us
return
to
the
case
of
an
arbitrary
stable
log
curve
X
log
.
Then
it
follows
immediately
from
the
description
of
the
relationship
between
the
abelianization
of
Π
1
and
the
abelianizations
of
verticial
subgroups
of
Π
1
given
in
[NodNon],
Lemma
1.4,
together
with
the
portion
of
assertion
(i)
that
has
already
been
verified,
that
α
stabilizes
and
induces
the
identity
automorphism
on
each
of
the
irreducible
components
of
X
log
of
genus
≥
1.
Next,
let
us
observe
that
it
follows
immediately
from
the
definition
of
l
aut
,
together
with
the
well-known
structure
of
the
submodule
of
the
abelianization
of
Π
1
generated
by
the
cuspidal
inertia
subgroups,
that
α
acts
trivially
on
the
set
of
cusps
of
X
log
.
Thus,
by
considering
the
various
connected
components
of
the
union
of
the
genus
zero
irreducible
components
of
X
log
,
we
conclude
that,
to
complete
the
verification
of
assertion
(i),
it
suffices
to
verify,
in
the
case
where
g
=
0,
that
any
automorphism
of
X
log
over
(Spec
k)
log
that
acts
trivially
on
the
set
of
cusps
of
X
log
is
equal
to
the
identity
automorphism.
But
this
follows
immediately
by
induction
on
r,
i.e.,
by
considering,
when
r
≥
4,
the
stable
log
curve
obtained
from
X
log
by
“forgetting”,
successively,
each
of
the
cusps
of
X
log
.
[Here,
we
apply
the
elementary
combinatorial
fact
that
every
non-smooth
pointed
stable
curve
of
genus
0
has
at
least
two
irreducible
components
that
contain
cusps.]
This
completes
the
proof
of
asser-
tion
(i).
Assertion
(ii)
follows
immediately
from
assertion
(i),
together
with
well-known
generalities
concerning
algebraic
stacks
[cf.,
e.g.,
the
discussion
surrounding
[FC],
Chapter
I,
Theorem
4.10].
54
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
3.
Applications
to
the
theory
of
tempered
fundamental
groups
In
the
present
§3,
we
apply
the
technical
tools
developed
in
the
pre-
ceding
§2,
together
with
the
theory
of
[CbTpI],
§5,
to
obtain
applica-
tions
to
the
theory
of
tempered
fundamental
groups.
In
particular,
we
prove
a
generalization
of
a
result
due
to
André
[cf.
[André],
Theorems
7.2.1,
7.2.3]
concerning
the
characterization
of
the
local
Galois
groups
in
the
image
of
the
outer
Galois
action
associated
to
a
hyperbolic
curve
over
a
number
field
[cf.
Corollary
3.20,
(iii),
below].
Definition
3.1.
Let
n
be
a
nonnegative
integer.
For
∈
{◦,
•},
let
p
be
a
prime
number;
Σ
a
nonempty
set
of
prime
numbers
such
that
Σ
=
{
p};
R
a
mixed
characteristic
complete
discrete
valuation
ring
of
residue
characteristic
p
whose
residue
field
is
separably
closed;
K
the
field
of
fractions
of
R;
K
an
algebraic
closure
of
K.
Write
def
I
K
=
Gal(
K/
K)
for
the
absolute
Galois
group
of
K;
R
for
the
∧
∧
ring
of
integers
of
K;
R
for
the
p-adic
completion
of
R;
K
for
∧
the
field
of
fractions
of
R
.
If
n
≥
2,
then
we
suppose
further
that
Σ
is
either
equal
to
Primes
or
of
cardinality
one.
Let
X
log
K
def
be
a
smooth
log
curve
over
K.
Write
X
log
K
=
X
log
K
×
K
K;
(X
K
)
log
n
for
the
n-th
log
configuration
space
[cf.
the
discussion
entitled
“Curves”
in
[CbTpII],
§0]
of
the
smooth
log
curve
X
log
K
over
K.
(i)
We
shall
write
def
Σ
Π
n
=
π
1
((X
K
)
log
n
)
for
the
maximal
pro-
Σ
quotient
of
the
log
fundamental
group
of
(X
K
)
log
n
.
Thus,
we
have
a
natural
outer
Galois
action
ρ
n
:
I
K
−→
Out(
Π
n
)
.
Note
that
Π
n
is
equipped
with
a
natural
structure
of
pro-
Σ
configuration
space
group
[cf.
[MzTa],
Definition
2.3,
(i)].
(ii)
We
shall
write
∧
π
1
temp
((X
K
)
log
n
×
K
K
)
∧
for
the
tempered
fundamental
group
of
(X
K
)
log
n
×
K
K
[cf.
[André],
§4].
[Here,
we
note
that
[André],
§4,
only
discusses
the
∧
case
where
the
base
field
K
is
a
complete
subfield
of
“C
p
”.
On
the
other
hand,
let
us
recall
from
[AbsTpI],
Proposition
2.2,
that
any
profinite
group
of
GFG-type
[cf.
[AbsTpI],
Definition
COMBINATORIAL
ANABELIAN
TOPICS
III
55
2.1,
(i)]
is
topologically
finitely
generated,
which
implies
that
the
set
of
open
subgroups
of
a
profinite
group
of
GFG-type
[such
as
Π
n
]
is
countable.
In
particular,
one
verifies
easily
[cf.
also
[Brk],
Corollary
9.5,
and
the
following
discussion]
that
the
construction
of
the
tempered
fundamental
group
given
in
∧
[André],
§4,
applies
even
in
the
case
where
the
base
field
K
is
not
a
complete
subfield
of
“C
p
”.]
We
shall
write
∧
Π
tp
=
lim
π
1
temp
((X
K
)
log
n
n
×
K
K
)/N
←−
def
N
∧
for
the
Σ-tempered
fundamental
group
of
(X
K
)
log
n
×
K
K
[cf.
[CmbGC],
Corollary
2.10,
(iii)],
i.e.,
the
inverse
limit
given
by
allowing
N
to
vary
over
the
open
normal
subgroups
of
∧
π
1
temp
((X
K
)
log
n
×
K
K
)
such
that
the
quotient
by
N
cor-
responds
to
a
topological
covering
[cf.
[André],
§4.2]
of
some
∧
finite
log
étale
Galois
covering
of
(X
K
)
log
n
×
K
K
of
degree
a
product
of
primes
∈
Σ.
[Here,
we
recall
that,
when
n
=
1,
such
a
“topological
covering”
corresponds
to
a
“combinatorial
covering”,
i.e.,
a
covering
determined
by
a
covering
of
the
dual
semi-graph
of
the
special
fiber
of
the
stable
model
of
some
fi-
∧
nite
log
étale
covering
of
(X
K
)
log
n
×
K
K
.]
Thus,
we
have
a
natural
outer
Galois
action
tp
ρ
n
:
I
K
−→
Out(
Π
tp
n
)
[cf.
[André],
Proposition
5.1.1].
Lemma
3.2
(Pro-Σ
completions
of
discrete
free
groups).
Let
Σ
be
a
nonempty
set
of
prime
numbers
and
F
a
discrete
free
group.
Then
the
following
hold:
(i)
The
natural
homomorphism
F
→
F
Σ
from
F
to
the
pro-Σ
completion
F
Σ
of
F
is
injective.
(ii)
Suppose
that
F
is
not
of
rank
one.
Then
the
image
of
the
injection
F
→
F
Σ
of
(i)
is
normally
terminal
[cf.
the
dis-
cussion
entitled
“Topological
groups”
in
[CbTpI],
§0].
Proof.
Assertion
(i)
follows
immediately
from
[RZ],
Proposition
3.3.15.
Assertion
(ii)
follows
immediately
from
the
fact
that
F
is
conjugacy
l-separable
for
every
prime
number
l
[cf.
[Prs],
Theorem
3.2],
together
with
a
similar
argument
to
the
argument
applied
in
the
proof
of
[André],
Lemma
3.2.1.
This
completes
the
proof
of
Lemma
3.2.
Proposition
3.3
(Log
and
tempered
fundamental
groups).
In
the
notation
of
Definition
3.1,
the
following
hold:
56
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Σ
(i)
Write
(
Π
tp
for
the
pro-
Σ
completion
of
Π
tp
n
)
n
.
Then
there
tp
Σ
∼
exists
a
natural
outer
isomorphism
(
Π
n
)
→
Π
n
.
(ii)
The
outer
homomorphism
Π
tp
1
→
Π
1
determined
by
the
outer
isomorphism
of
(i)
is
injective.
(iii)
The
image
of
the
outer
injection
Π
tp
1
→
Π
1
of
(ii)
is
nor-
mally
terminal.
•
tp
◦
•
(iv)
Write
Isom(
◦
Π
tp
1
,
Π
1
)
(respectively,
Isom(
Π
1
,
Π
1
))
for
the
◦
•
tp
set
of
isomorphisms
of
◦
Π
tp
1
(respectively,
Π
1
)
with
Π
1
(re-
spectively,
•
Π
1
)
and
Inn(−)
for
the
group
of
inner
automor-
phisms
of
“(−)”.
Then
the
natural
map
between
sets
of
outer
isomorphisms
[i.e.,
sets
of
“Inn(−)-orbits”]
•
tp
•
tp
◦
•
•
Isom(
◦
Π
tp
1
,
Π
1
)/Inn(
Π
1
)
−→
Isom(
Π
1
,
Π
1
)/Inn(
Π
1
)
induced
by
the
natural
outer
isomorphism
of
(i)
—
hence
also
the
natural
homomorphism
Out(
Π
tp
1
)
−→
Out(
Π
1
)
—
is
injective.
Proof.
Assertion
(i)
follows
immediately
from
the
various
definitions
involved.
Next,
we
verify
assertion
(ii)
(respectively,
(iii)).
Let
us
first
observe
that
it
follows
immediately
from
assertion
(i)
that,
to
verify
assertion
(ii)
(respectively,
(iii)),
by
replacing
X
log
K
by
a
suitable
con-
nected
finite
log
étale
covering
of
X
log
K
,
we
may
assume
without
loss
of
generality
that
the
first
Betti
number
of
the
dual
semi-graph
of
the
spe-
cial
fiber
of
the
stable
model
of
every
connected
finite
log
étale
covering
of
X
log
K
is
=
1.
Then
since
Π
tp
1
is
a
projective
limit
of
extensions
of
finite
groups
whose
orders
are
products
of
primes
∈
Σ
by
discrete
free
groups
whose
ranks
are
=
1,
assertion
(ii)
(respectively,
(iii))
follows
immediately
from
Lemma
3.2,
(i)
(respectively,
(ii)).
This
completes
the
proof
of
assertion
(ii)
(respectively,
(iii)).
Assertion
(iv)
follows
immediately
from
assertion
(iii).
This
completes
the
proof
of
Proposi-
tion
3.3.
Remark
3.3.1.
The
injections
of
Proposition
3.3,
(iv),
allow
one
to
•
tp
•
tp
tp
regard
Isom(
◦
Π
tp
1
,
Π
1
)/Inn(
Π
1
),
(respectively,
Out(
Π
1
))
as
a
sub-
set
(respectively,
subgroup)
of
Isom(
◦
Π
1
,
•
Π
1
)/Inn(
•
Π
1
)
(respectively,
Out(
Π
1
)).
Remark
3.3.2.
The
normal
terminality
of
Proposition
3.3,
(iii),
may
also
be
verified
by
applying
the
theory
of
[SemiAn]
and
[NodNon].
We
refer
to
the
proof
of
[IUTeichI],
Proposition
2.4,
(iii),
for
more
details
concerning
this
approach.
COMBINATORIAL
ANABELIAN
TOPICS
III
57
Definition
3.4.
Let
G
be
a
[semi-]graph.
Write
Node(G)
for
the
set
of
closed
edges
of
G.
Then
we
shall
refer
to
a
map
def
μ
:
Node(G)
→
R
>0
=
{
a
∈
R
|
a
>
0
}
as
a
metric
structure
on
G.
Also,
we
shall
refer
to
a
[semi-]graph
equipped
with
a
metric
structure
as
a
metric
[semi-]graph.
Let
Σ
be
a
[possibly
empty]
set
of
prime
numbers.
Then
we
shall
say
that
an
∼
isomorphism
G
1
→
G
2
between
two
[semi-]graphs
G
1
,
G
2
equipped
with
metric
structures
μ
1
,
μ
2
is
Σ-rationally
compatible
with
the
given
metric
structures
if
there
exists
an
element
def
ξ
∈
(
Z
Σ
)
+
(⊆
Q
>0
=
Q
∩
R
>0
)
—
i.e.,
a
positive
rational
number
that
is
invertible,
as
an
integer,
at
the
primes
of
Σ
[cf.
the
notation
of
[CbTpI],
Corollary
5.9,
(iv),
if
def
Σ
=
∅;
set
(
Z
Σ
)
+
=
Q
>0
if
Σ
=
∅]
—
such
that
ξ
·
μ
1
coincides
with
the
∼
composite
of
the
bijection
Node(G
1
)
→
Node(G
2
)
determined
by
the
given
isomorphism
with
μ
2
.
[Thus,
if
G
1
=
G
2
is
a
finite
[semi-]graph,
and
μ
1
=
μ
2
,
then
such
a
ξ
is
necessarily
equal
to
1.
Alternatively,
if
Σ
=
Primes,
then
such
a
ξ
is
necessarily
equal
to
1.]
Definition
3.5.
In
the
notation
of
Definition
3.1,
let
Σ
⊆
Σ
\
{
p}
be
a
nonempty
subset
of
Σ
\
{
p}
and
H
⊆
Π
1
an
open
subgroup
of
Π
1
.
(i)
We
shall
write
G
H
[Σ]
for
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
deter-
mined
by
the
special
fiber
[cf.
[CmbGC],
Example
2.5]
of
the
stable
model
over
R
of
the
connected
finite
log
étale
covering
of
X
log
K
corresponding
to
H
⊆
Π
1
.
(ii)
We
shall
write
G
H
for
the
semi-graph
associated
to
[i.e.,
the
dual
semi-graph
of]
the
special
fiber
of
the
stable
model
over
R
of
the
connected
finite
log
étale
covering
of
X
log
K
corresponding
to
H
⊆
Π
1
,
i.e.,
the
underlying
semi-graph
of
G
H
[Σ]
[cf.
(i)].
Note
that
this
semi-graph
is
independent
of
the
choice
of
Σ.
(iii)
We
shall
write
μ
H
:
Node(G
H
)
−→
R
>0
for
the
metric
structure
[cf.
Definition
3.4]
on
G
H
associated
to
the
stable
model
over
R
of
the
connected
finite
log
étale
58
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
covering
of
X
log
K
corresponding
to
H
⊆
Π
1
,
i.e.,
the
metric
structure
defined
as
follows:
∧
Write
v
K
∧
for
the
p-adic
valuation
of
K
such
that
v
K
∧
(
p)
=
1.
Let
e
∈
Node(G
H
).
Suppose
∧
that
the
R
-algebra
given
by
the
completion
at
the
node
corresponding
to
e
of
the
stable
model
of
the
connected
covering
of
X
log
K
determined
by
H
⊆
Π
1
is
isomorphic
to
∧
R
[[s
1
,
s
2
]]/(s
1
s
2
−
a
e
)
∧
—
where
a
e
∈
R
is
a
nonzero
non-unit,
and
s
1
and
def
s
2
denote
indeterminates.
Then
we
set
μ
H
(e)
=
v
K
∧
(a
e
):
μ
H
:
Node(G
H
)
−→
R
>0
e
→
v
K
∧
(a
e
)
.
Here,
one
verifies
easily
that
“μ
H
(a
e
)”
depends
only
on
e,
i.e.,
is
independent
of
the
choice
of
the
local
equation
“s
1
s
2
−
a
e
”.
Remark
3.5.1.
In
the
notation
of
Definition
3.5,
it
follows
immedi-
ately
from
the
various
definitions
involved
that
one
has
a
natural
outer
isomorphism
∼
(
H)
Σ
−→
Π
G
H
[Σ]
between
the
maximal
pro-Σ
quotient
(
H)
Σ
of
H
and
the
[pro-Σ]
fundamental
group
Π
G
H
[Σ]
of
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
G
H
[Σ].
Proposition
3.6
(Equivalences
of
properties
of
isomorphisms
between
fundamental
groups).
In
the
notation
of
Definition
3.1,
∼
let
α
:
◦
Π
1
→
•
Π
1
be
an
isomorphism
of
profinite
groups.
[Thus,
it
follows
immediately
that
◦
Σ
=
•
Σ
—
cf.,
e.g.,
the
proof
of
[CbTpI],
Proposition
1.5,
(i).]
Consider
the
following
conditions:
∼
(a)
The
outer
isomorphism
◦
Π
1
→
•
Π
1
determined
by
α
is
con-
tained
in
•
tp
•
tp
◦
•
•
Isom(
◦
Π
tp
1
,
Π
1
)/Inn(
Π
1
)
⊆
Isom(
Π
1
,
Π
1
)/Inn(
Π
1
)
[cf.
Remark
3.3.1],
and
◦
Σ
=
•
Σ
⊆
{
◦
p,
•
p}.
(b
∀
)
For
any
characteristic
open
subgroup
◦
H
⊆
◦
Π
1
of
◦
Π
1
and
any
nonempty
subset
Σ
⊆
◦
Σ
=
•
Σ
such
that
◦
p,
•
p
∈
Σ,
if
def
we
write
•
H
=
α(
◦
H)
⊆
•
Π
1
,
then
the
outer
isomorphism
∼
∼
of
(
◦
H)
Σ
→
Π
G
◦
H
[Σ]
[cf.
Remark
3.5.1]
with
(
•
H)
Σ
→
Π
G
•
H
[Σ]
COMBINATORIAL
ANABELIAN
TOPICS
III
59
induced
by
α
is
group-theoretically
verticial
[cf.
[CmbGC],
Definition
1.4,
(iv)].
(b
∃
)
For
any
characteristic
open
subgroup
◦
H
⊆
◦
Π
1
of
◦
Π
1
,
there
exists
a
nonempty
subset
Σ
⊆
◦
Σ
=
•
Σ
[which
may
depend
on
def
◦
H]
such
that
◦
p,
•
p
∈
Σ,
and,
moreover,
if
we
write
•
H
=
∼
α(
◦
H)
⊆
•
Π
1
,
then
the
outer
isomorphism
of
(
◦
H)
Σ
→
Π
G
◦
H
[Σ]
∼
[cf.
Remark
3.5.1]
with
(
•
H)
Σ
→
Π
G
•
H
[Σ]
induced
by
α
is
group-theoretically
verticial.
(c
∀
)
For
any
characteristic
open
subgroup
◦
H
⊆
◦
Π
1
of
◦
Π
1
and
any
nonempty
subset
Σ
⊆
◦
Σ
=
•
Σ
such
that
◦
p,
•
p
∈
Σ,
if
def
we
write
•
H
=
α(
◦
H)
⊆
•
Π
1
,
then
the
outer
isomorphism
of
∼
∼
(
◦
H)
Σ
→
Π
G
◦
H
[Σ]
[cf.
Remark
3.5.1]
with
(
•
H)
Σ
→
Π
G
•
H
[Σ]
induced
by
α
is
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)].
(c
∃
)
For
any
characteristic
open
subgroup
◦
H
⊆
◦
Π
1
of
◦
Π
1
,
there
exists
a
nonempty
subset
Σ
⊆
◦
Σ
=
•
Σ
[which
may
depend
on
def
◦
H]
such
that
◦
p,
•
p
∈
Σ,
and,
moreover,
if
we
write
•
H
=
∼
α(
◦
H)
⊆
•
Π
1
,
then
the
outer
isomorphism
of
(
◦
H)
Σ
→
Π
G
◦
H
[Σ]
∼
[cf.
Remark
3.5.1]
with
(
•
H)
Σ
→
Π
G
•
H
[Σ]
induced
by
α
is
graphic.
Then:
(i)
We
have
implications:
(b
∀
)
⇐=
(c
∀
)
⇐=
(c
∃
)
=⇒
(a)
⇐⇒
(b
∃
)
=⇒
(b
∀
)
.
(ii)
Suppose
that
◦
Σ
=
•
Σ
⊆
{
◦
p,
•
p}.
[This
condition
is
satisfied
if,
for
instance,
◦
p
=
•
p.]
Then
we
have
equivalences:
(b
∃
)
⇐⇒
(b
∀
)
and
(c
∃
)
⇐⇒
(c
∀
)
.
(iii)
Suppose
that
either
◦
p
∈
◦
Σ
or
•
p
∈
•
Σ.
Then
we
have
equiva-
lences:
(a)
⇐⇒
(b
∃
)
⇐⇒
(c
∃
)
.
Moreover,
(a),
(b
∃
),
and
(c
∃
)
imply
that
◦
p
=
•
p.
Proof.
First,
we
claim
that
the
following
assertion
holds:
Claim
3.6.A:
Suppose
that
(a)
is
satisfied,
and
that
either
◦
p
∈
◦
Σ
or
•
p
∈
•
Σ.
Then
◦
p
=
•
p
∈
◦
Σ
=
•
Σ.
Moreover,
(c
∃
)
is
satisfied.
To
verify
Claim
3.6.A,
suppose
that
(a)
is
satisfied,
and
that
◦
p
∈
◦
Σ.
Then
it
follows
immediately
from
[SemiAn],
Corollary
3.11
[cf.,
espe-
cially,
the
portion
of
the
statement
and
proof
of
[SemiAn],
Corollary
3.11,
concerning,
in
the
notation
of
loc.
cit.,
the
assertion
“p
α
=
p
β
”];
60
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[SemiAn],
Remark
3.11.1
[cf.
also
[AbsTpII],
Lemma
2.6,
(i);
the
state-
ment
and
proof
of
[AbsTpII],
Corollary
2.11;
[AbsTpII],
Remark
2.11.1,
(i)],
that
◦
p
=
•
p
∈
◦
Σ
=
•
Σ,
and,
moreover,
that
(c
∃
)
is
satisfied.
This
completes
the
proof
of
Claim
3.6.A.
Next,
we
verify
assertion
(i).
Let
us
first
observe
that
it
follows
from
the
fact
that
graphicity
implies
group-theoretic
verticiality
that
the
following
implications
hold:
(c
∀
)
⇒
(b
∀
)
and
(c
∃
)
⇒
(b
∃
).
Next,
we
verify
the
implication
(b
∃
)
⇒
(b
∀
)
(respectively,
(c
∃
)
⇒
(c
∀
)).
Suppose
that
(b
∃
)
(respectively,
(c
∃
))
is
satisfied.
Then
it
follows
that
◦
Σ
=
•
Σ
⊆
{
◦
p,
•
p}.
Next,
let
us
observe
that,
to
complete
the
verification
of
(b
∀
)
(respectively,
(c
∀
)),
we
may
assume
without
loss
of
generality
—
by
replacing
the
open
subgroup
◦
H
⊆
◦
Π
1
in
(b
∀
)
(respectively,
(c
∀
))
by
◦
Π
1
—
that
◦
H
=
◦
Π
1
and
•
H
=
•
Π
1
.
Moreover,
one
verifies
easily
that,
to
complete
the
verification
of
(b
∀
)
(respectively,
(c
∀
)),
we
may
assume
without
loss
of
generality
—
by
replacing
the
subset
Σ
in
(b
∀
)
(respectively,
(c
∀
))
by
◦
Σ\(
◦
Σ∩{
◦
p,
•
p})
=
•
Σ\(
•
Σ∩{
◦
p,
•
p})
(
=
∅)
—
that
Σ
=
◦
Σ
\
(
◦
Σ
∩
{
◦
p,
•
p})
=
•
Σ
\
(
•
Σ
∩
{
◦
p,
•
p})
(
=
∅).
Let
◦
U
⊆
◦
Π
1
def
be
a
characteristic
open
subgroup.
Write
•
U
=
α(
◦
U
)
⊆
•
Π
1
.
Then
it
follows
immediately
from
(b
∃
)
(respectively,
(c
∃
))
that
there
exists
a
nonempty
subset
Σ
◦
U
⊆
Σ
such
that
α
induces
a
functorial
bijection
∼
Vert(G
◦
U
[Σ])
=
Vert(G
◦
U
[Σ
◦
U
])
−→
Vert(G
•
U
[Σ
◦
U
])
=
Vert(G
•
U
[Σ])
(respectively,
∼
VCN(G
◦
U
[Σ])
=
VCN(G
◦
U
[Σ
◦
U
])
−→
VCN(G
•
U
[Σ
◦
U
])
=
VCN(G
•
U
[Σ])).
In
particular,
by
considering
these
functorial
bijections
between
the
sets
“Vert”
(respectively,
“VCN”)
associated
to
the
connected
finite
étale
coverings
corresponding
to
the
various
characteristic
open
sub-
def
groups
◦
U
⊆
◦
Π
1
,
•
U
=
α(
◦
U
)
⊆
•
Π
1
,
we
conclude
that
the
isomor-
∼
•
Σ
phism
◦
Π
Σ
1
→
Π
1
is
group-theoretically
verticial
(respectively,
group-
theoretically
verticial
and
group-theoretically
edge-like,
hence
graphic
[cf.
[CmbGC],
Proposition
1.5,
(ii)]).
This
completes
the
proof
of
the
implication
(b
∃
)
⇒
(b
∀
)
(respectively,
(c
∃
)
⇒
(c
∀
)).
Next,
we
observe
that
since
(a)
implies
that
◦
Σ
=
•
Σ
⊆
{
◦
p,
•
p},
the
implication
(a)
⇒
(b
∃
)
follows
from
[SemiAn],
Theorem
3.7,
(iv),
together
with
[the
evident
Σ-tempered
analogue
of]
the
discussion
of
[SemiAn],
Example
2.10.
Thus,
to
complete
the
verification
of
assertion
(i),
it
suffices
to
verify
the
implication
(b
∃
)
⇒
(a).
To
this
end,
suppose
that
(b
∃
)
is
satisfied.
Let
◦
H
⊆
◦
Π
1
be
a
characteristic
open
subgroup
of
◦
Π
1
.
Then
it
follows
from
(b
∃
)
that
there
exists
a
nonempty
subset
def
Σ
⊆
◦
Σ
=
•
Σ
such
that
◦
p,
•
p
∈
Σ,
and,
moreover,
if
we
write
•
H
=
∼
α(
◦
H)
⊆
•
Π
1
,
then
the
outer
isomorphism
of
(
◦
H)
Σ
→
Π
G
◦
H
[Σ]
[cf.
∼
Remark
3.5.1]
with
(
•
H)
Σ
→
Π
G
•
H
[Σ]
induced
by
α
is
group-theoretically
COMBINATORIAL
ANABELIAN
TOPICS
III
61
verticial.
For
each
∈
{◦,
•},
write
G
=c
H
[Σ]
for
the
graph
of
anabelioids
obtained
by
omitting
the
cusps
[i.e.,
open
edges]
of
G
H
[Σ];
tp
=c
Π
tp
G
[Σ]
,
Π
G
[Σ]
H
H
for
the
tempered
fundamental
groups
of
G
H
[Σ],
G
=c
H
[Σ],
respectively
[cf.
the
discussion
preceding
[SemiAn],
Proposition
3.6].
Here,
let
us
observe
that
it
follows
immediately
from
the
various
definitions
involved
that
we
have
a
natural
commutative
diagram
∼
=c
Π
tp
G
[Σ]
−→
H
∩
Π
tp
G
[Σ]
H
∩
∼
∼
=c
Σ
Σ
(Π
tp
−→
(Π
tp
−→
Π
G
H
[Σ]
G
[Σ]
)
G
[Σ]
)
H
H
=c
tp
Σ
Σ
—
where
we
write
(Π
tp
G
[Σ]
)
,
(Π
G
[Σ]
)
for
the
pro-Σ
completions
of
H
H
=c
tp
Π
tp
G
H
[Σ]
,
Π
G
H
[Σ]
,
respectively;
the
horizontal
arrows
are
outer
isomor-
phisms;
the
lower
right-hand
horizontal
arrow
is
the
outer
isomorphism
of
Proposition
3.3,
(i);
the
vertical
inclusions
are
the
inclusions
that
arise
from
Proposition
3.3,
(ii).
∼
∼
Now
since
the
outer
isomorphism
of
(
◦
H)
Σ
→
Π
G
◦
H
[Σ]
with
(
•
H)
Σ
→
Π
G
•
H
[Σ]
induced
by
α
is
group-theoretically
verticial,
it
follows
imme-
diately
from
[NodNon],
Proposition
1.13;
the
argument
applied
in
the
proof
of
the
sufficiency
portion
of
[CmbGC],
Proposition
1.5,
(ii),
that
∼
=c
α
determines
an
isomorphism
G
◦
=c
H
[Σ]
→
G
•
H
[Σ]
of
graphs
of
anabe-
lioids.
Thus,
it
follows
immediately
from
the
existence
of
the
natural
outer
isomorphisms
discussed
above
that
the
[group-theoretically
verti-
∼
cial]
outer
isomorphism
Π
G
◦
H
[Σ]
→
Π
G
•
H
[Σ]
induced
by
the
isomorphism
α
maps
the
Π
G
◦
H
[Σ]
-conjugacy
class
of
Π
tp
G
◦
H
[Σ]
to
the
Π
G
•
H
[Σ]
-conjugacy
tp
class
of
Π
G
•
H
[Σ]
.
Moreover,
it
follows
immediately
from
the
normal
terminality
of
Proposition
3.3,
(iii),
that
the
resulting
conjugacy
inde-
terminacies
may
be
reduced
to
Π
tp
G
H
[Σ]
-conjugacy
indeterminacies.
In
particular,
by
applying
these
observations
to
the
various
characteristic
open
subgroups
“
◦
H”
of
◦
Π
1
,
one
verifies
easily
from
the
description
of
the
tempered
fundamental
group
as
a
[countably
indexed!]
projective
limit
given
in
[André],
§4.5
[cf.
also
the
discussion
preceding
[SemiAn],
Proposition
3.6,
as
well
as
the
discussion
of
Definition
3.1,
(ii),
of
the
∼
present
paper],
that
the
outer
isomorphism
◦
Π
1
→
◦
Π
1
determined
by
α
•
tp
•
tp
◦
•
•
is
contained
in
Isom(
◦
Π
tp
1
,
Π
1
)/Inn(
Π
1
)
⊆
Isom(
Π
1
,
Π
1
)/Inn(
Π
1
),
i.e.,
that
(a)
is
satisfied.
This
completes
the
proof
of
the
implication
62
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(b
∃
)
⇒
(a),
hence
also
of
assertion
(i).
Assertion
(ii)
follows
immedi-
ately
from
assertion
(i),
together
with
the
various
definitions
involved.
Assertion
(iii)
follows
from
assertion
(i),
together
with
Claim
3.6.A.
This
completes
the
proof
of
Proposition
3.6.
Definition
3.7.
In
the
notation
of
Definition
3.1:
∼
(i)
Let
α
:
◦
Π
1
→
•
Π
1
be
an
isomorphism
of
profinite
groups.
Then
we
shall
say
that
α
is
G-admissible
[i.e.,
“graph-admissible”]
if
α
satisfies
condition
(c
∃
)
—
hence
also
conditions
(a),
(b
∀
),
(b
∃
),
(c
∀
)
[cf.
Proposition
3.6,
(i)]
—
of
Proposition
3.6.
Write
Aut(
◦
Π
1
)
G
⊆
Aut(
◦
Π
1
)
for
the
subgroup
[cf.
the
equivalence
(c
∀
)
⇔
(c
∃
)
of
Proposi-
tion
3.6,
(ii))]
of
G-admissible
automorphisms
of
◦
Π
1
and
Out(
◦
Π
1
)
G
=
Aut(
◦
Π
1
)
G
/Inn(
◦
Π
1
)
⊆
Out(
◦
Π
1
)
def
for
the
subgroup
of
G-admissible
outomorphisms
of
◦
Π
1
.
∼
(ii)
Let
α
:
◦
Π
1
→
•
Π
1
be
an
isomorphism
of
profinite
groups
[so
◦
Σ
=
•
Σ
—
cf.,
e.g.,
the
proof
of
[CbTpI],
Proposition
1.5,
(i)].
Let
Σ
⊆
◦
Σ
=
•
Σ
be
a
[possibly
empty]
subset
such
that
◦
p,
•
p
∈
Σ.
Then
we
shall
say
that
α
is
Σ-M-admissible
[i.e.,
“Σ-metric-admissible”]
if
α
is
G-admissible
[cf.
(i)],
and,
moreover,
the
following
condition
is
satisfied:
Let
◦
H
⊆
◦
Π
1
be
a
characteristic
open
subgroup
of
def
◦
Π
1
.
Write
•
H
=
α(
◦
H)
⊆
•
Π
1
.
Then
the
isomor-
phism
of
G
◦
H
with
G
•
H
induced
by
α
[where
we
note
that
one
verifies
easily
that
the
isomorphism
of
G
◦
H
with
G
•
H
induced
by
α
does
not
depend
on
the
choice
of
“Σ”
in
condition
(c
∀
)
of
Proposition
3.6]
is
Σ-
rationally
compatible
[cf.
Definition
3.4]
with
respect
to
the
metric
structures
μ
◦
H
,
μ
•
H
[cf.
Definition
3.5,
(iii)].
[Thus,
if
the
collections
of
data
labeled
by
◦,
•
are
equal,
then
the
notion
of
Σ-M-admissibility
is
independent
of
the
choice
of
Σ
—
cf.
the
final
portion
of
Definition
3.4.]
We
shall
say
that
α
is
M-admissible
if
α
is
∅-M-admissible.
Write
Aut(
◦
Π
1
)
M
⊆
Aut(
◦
Π
1
)
for
the
subgroup
of
M-admissible
automorphisms
of
◦
Π
1
and
Out(
◦
Π
1
)
M
=
Aut(
◦
Π
1
)
M
/Inn(
◦
Π
1
)
⊆
Out(
◦
Π
1
)
def
for
the
subgroup
of
M-admissible
outomorphisms
of
◦
Π
1
.
COMBINATORIAL
ANABELIAN
TOPICS
III
63
(iii)
We
shall
write
Out
F
(
◦
Π
n
)
M
⊆
Out
F
(
◦
Π
n
)
for
the
subgroup
of
the
group
Out
F
(
◦
Π
n
)
of
F-admissible
outo-
morphisms
of
◦
Π
n
[cf.
[CmbCsp],
Definition
1.1,
(ii)]
obtained
by
forming
the
inverse
image
of
Out(
◦
Π
1
)
M
⊆
Out(
◦
Π
1
)
[cf.
(ii)]
via
the
natural
homomorphism
Out
F
(
◦
Π
n
)
→
Out
F
(
◦
Π
1
)
=
Out(
◦
Π
1
)
[cf.
[CbTpI],
Theorem
A,
(i)];
Out
FC
(
◦
Π
n
)
M
=
Out
F
(
◦
Π
n
)
M
∩
Out
C
(
◦
Π
n
)
⊆
Out
FC
(
◦
Π
n
)
def
[cf.
[CmbCsp],
Definition
1.1,
(ii)].
Definition
3.8.
In
the
notation
of
Definition
3.1:
∼
(i)
Let
α
:
◦
Π
n
→
•
Π
n
be
an
isomorphism
of
profinite
groups
[so
◦
Σ
=
•
Σ
—
cf.,
e.g.,
the
proof
of
[CbTpI],
Proposition
1.5,
(i)]
and
l
∈
◦
Σ
=
•
Σ
such
that
l
∈
{
◦
p,
•
p}.
Then
we
shall
say
that
α
is
{l}-I-admissible
[i.e.,
“{l}-inertia-admissible”]
if
α
is
PF-admissible
whenever
n
≥
2
[cf.
[CbTpI],
Definition
1.4,
(i)],
and,
moreover,
the
following
condition
is
satisfied:
Let
◦
Π
n
(
◦
Π
n
)
∗
be
an
F-characteristic
almost
pro-l
quotient
of
◦
Π
n
(
π
1
((X
◦
K
)
log
n
))
[cf.
Definition
2.1,
(iii)].
If
◦
Σ
=
•
Σ
=
Primes,
then
we
assume
fur-
ther
that
the
quotient
◦
Π
n
(
◦
Π
n
)
∗
is
an
almost
maximal
pro-l
quotient
relative
to
some
characteris-
tic
open
subgroup
of
◦
Π
n
[cf.
Definition
1.1].
Write
•
Π
n
(
•
Π
n
)
∗
for
the
quotient
of
•
Π
n
that
corre-
sponds
to
◦
Π
n
(
◦
Π
n
)
∗
via
α.
[Here,
we
observe
that
since
α
is
PF-admissible
whenever
n
≥
2,
one
verifies
immediately
that
the
quotient
•
Π
n
(
•
Π
n
)
∗
satisfies
similar
assumptions
to
the
assumptions
im-
posed
on
the
quotient
◦
Π
n
(
◦
Π
n
)
∗
.]
Then
there
exist
open
subgroups
◦
J
⊆
I
◦
K
,
•
J
⊆
I
•
K
[which
may
depend
on
◦
Π
n
(
◦
Π
n
)
∗
]
such
that
the
diagram
Im(
◦
J)
−−−→
Out((
◦
Π
n
)
∗
)
⏐
⏐
⏐
⏐
β
Im(
•
J)
−−−→
Out((
•
Π
n
)
∗
)
—
where,
for
∈
{◦,
•},
we
write
Im(
J)
⊆
Out((
Π
n
)
∗
)
for
the
image
of
J
via
the
homomorphism
J
→
Out((
Π
n
)
∗
)
induced
[in
light
of
our
assumptions
on
64
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
the
quotients
under
consideration!]
by
ρ
n
;
the
hor-
izontal
arrows
are
the
natural
inclusions;
the
right-
hand
vertical
arrow
is
the
isomorphism
induced
by
the
isomorphism
α
—
commutes
for
some
[uniquely
∼
determined]
isomorphism
β
:
Im(
◦
J)
→
Im(
•
J).
∼
We
shall
say
that
an
outer
isomorphism
◦
Π
n
→
•
Π
n
is
{l}-I-
∼
admissible
if
it
arises
from
an
isomorphism
◦
Π
n
→
•
Π
n
which
is
{l}-I-admissible.
∼
(ii)
We
shall
say
that
an
isomorphism
of
profinite
groups
◦
Π
n
→
•
Π
n
[so
◦
Σ
=
•
Σ
—
cf.,
e.g.,
the
proof
of
[CbTpI],
Proposition
1.5,
(i)]
is
I-admissible
[i.e.,
“inertia-admissible”]
if
◦
Σ
=
•
Σ
⊆
{
◦
p,
•
p},
and,
moreover,
the
isomorphism
is
{l}-I-admissible
[cf.
(i)]
for
every
prime
number
l
∈
◦
Σ
=
•
Σ
such
that
l
∈
∼
{
◦
p,
•
p}.
We
shall
say
that
an
outer
isomorphism
◦
Π
n
→
•
Π
n
∼
is
I-admissible
if
it
arises
from
an
isomorphism
◦
Π
n
→
•
Π
n
which
is
I-admissible.
(iii)
Let
l
∈
◦
Σ
be
such
that
l
=
◦
p.
Then
we
shall
write
Aut
{l}-I
(
◦
Π
n
)
⊆
Aut(
◦
Π
n
)
for
the
subgroup
of
{l}-I-admissible
automorphisms
of
◦
Π
n
[cf.
(i)];
Out
{l}-I
(
◦
Π
n
)
=
Aut
{l}-I
(
◦
Π
n
)/Inn(
◦
Π
n
)
⊆
Out(
◦
Π
n
)
def
for
the
subgroup
of
{l}-I-admissible
outomorphisms
of
◦
Π
n
;
Out
F
{l}-I
(
◦
Π
n
)
=
Out
{l}-I
(
◦
Π
n
)
∩
Out
F
(
◦
Π
n
)
⊆
Out
F
(
◦
Π
n
)
def
[cf.
[CmbCsp],
Definition
1.1,
(ii)];
Out
FC
{l}-I
(
◦
Π
n
)
=
Out
{l}-I
(
◦
Π
n
)
∩
Out
FC
(
◦
Π
n
)
⊆
Out
FC
(
◦
Π
n
)
def
[cf.
[CmbCsp],
Definition
1.1,
(ii)].
Also,
we
shall
write
def
Aut
{l}-I
(
◦
Π
n
)
⊆
Aut(
◦
Π
n
)
Aut
I
(
◦
Π
n
)
=
l∈
◦
Σ\(
◦
Σ∩{
◦
p})
for
the
subgroup
of
I-admissible
automorphisms
of
◦
Π
n
[cf.
(ii)];
def
Out
I
(
◦
Π
n
)
=
Out
{l}-I
(
◦
Π
n
)
⊆
Out(
◦
Π
n
)
l∈
◦
Σ\(
◦
Σ∩{
◦
p})
for
the
subgroup
of
I-admissible
outomorphisms
of
◦
Π
n
;
Out
FI
(
◦
Π
n
)
=
Out
I
(
◦
Π
n
)
∩
Out
F
(
◦
Π
n
)
⊆
Out
F
(
◦
Π
n
)
;
def
Out
FCI
(
◦
Π
n
)
=
Out
I
(
◦
Π
n
)
∩
Out
FC
(
◦
Π
n
)
⊆
Out
FC
(
◦
Π
n
)
.
def
COMBINATORIAL
ANABELIAN
TOPICS
III
65
(iv)
Let
l
∈
◦
Σ
be
such
that
l
=
◦
p.
Then
we
shall
write
Out
F
(
◦
Π
n
)
{l}-I
⊆
Out
F
(
◦
Π
n
)
for
the
subgroup
of
the
group
Out
F
(
◦
Π
n
)
of
F-admissible
out-
omorphisms
of
◦
Π
n
obtained
by
forming
the
inverse
image
of
Out
{l}-I
(
◦
Π
1
)
⊆
Out(
◦
Π
1
)
[cf.
(iii)]
via
the
natural
homomor-
phism
Out
F
(
◦
Π
n
)
→
Out
F
(
◦
Π
1
)
=
Out(
◦
Π
1
)
[cf.
[CbTpI],
The-
orem
A,
(i)];
Out
FC
(
◦
Π
n
)
{l}-I
=
Out
F
(
◦
Π
n
)
{l}-I
∩
Out
C
(
◦
Π
n
)
⊆
Out
FC
(
◦
Π
n
)
.
def
Also,
we
shall
write
Out
F
(
◦
Π
n
)
I
=
def
Out
F
(
◦
Π
n
)
{l}-I
⊆
Out
F
(
◦
Π
n
)
;
l∈
◦
Σ\(
◦
Σ∩{
◦
p})
Out
FC
(
◦
Π
n
)
I
=
Out
F
(
◦
Π
n
)
I
∩
Out
C
(
◦
Π
n
)
⊆
Out
FC
(
◦
Π
n
)
.
def
Theorem
3.9
(Equivalence
of
metric-admissibility
and
inerti-
a-admissibility).
For
∈
{◦,
•},
let
p
be
a
prime
number;
Σ
a
nonempty
set
of
prime
numbers
such
that
Σ
=
{
p};
R
a
mixed
characteristic
complete
discrete
valuation
ring
of
residue
characteristic
p
whose
residue
field
is
separably
closed;
K
the
field
of
fractions
of
R;
K
an
algebraic
closure
of
K;
X
log
K
a
smooth
log
curve
over
K.
For
∈
{◦,
•},
write
X
log
K
=
X
log
K
×
K
K
;
def
def
Π
1
=
π
1
(X
log
K
)
Σ
for
the
maximal
pro-
Σ
quotient
of
the
log
fundamental
group
of
X
log
K
.
Let
∼
α
:
◦
Π
1
→
•
Π
1
be
an
isomorphism
of
profinite
groups.
[Thus,
it
follows
immediately
that
◦
Σ
=
•
Σ
—
cf.,
e.g.,
the
proof
of
[CbTpI],
Proposition
1.5,
(i).]
If
◦
p
∈
◦
Σ
and
•
p
∈
•
Σ,
then
we
assume
further
that
α
is
group-
theoretically
cuspidal
[cf.
[CmbGC],
Definition
1.4,
(iv)].
Then
the
following
conditions
are
equivalent:
(a)
α
is
M-admissible
[cf.
Definition
3.7,
(ii)].
(b
∀
)
α
is
I-admissible
[cf.
Definition
3.8,
(ii)].
(b
∃
)
There
exists
a
prime
number
l
∈
◦
Σ
=
•
Σ
such
that
l
∈
{
◦
p,
•
p},
and,
moreover,
α
is
{l}-I-admissible
[cf.
Defini-
tion
3.8,
(i)].
66
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Proof.
First,
let
us
observe
that
it
follows
formally
from
the
various
definitions
involved
[cf.
Definitions
3.7,
(i),
(ii);
3.8,
(ii)]
that
conditions
(a),
(b
∀
),
and
(b
∃
)
all
imply
that
there
exists
a
prime
number
l
∈
◦
Σ
=
•
Σ
such
that
l
∈
{
◦
p,
•
p}.
Now
fix
such
a
prime
number
l
and
consider
the
condition:
(b
{l}
):
α
is
{l}-I-admissible
[cf.
Definition
3.8,
(i)].
Then
[since
l
is
arbitrary,
and
condition
(a)
is
manifestly
independent
of
the
choice
of
l]
it
follows
formally
from
the
various
definitions
involved
that
to
verify
Theorem
3.9,
it
suffices
to
verify
the
equivalence
(a)
⇐⇒
(b
{l}
)
.
To
this
end,
let
◦
H
⊆
◦
Π
1
be
a
characteristic
open
subgroup
of
◦
Π
1
.
def
Write
•
H
=
α(
◦
H)
⊆
•
Π
1
.
Also,
for
each
∈
{◦,
•},
write
(
Π
1
)
∗
for
the
maximal
almost
pro-l
quotient
of
Π
1
with
respect
to
H.
[Thus,
∼
(
H)
{l}
→
Π
G
H
[{l}]
⊆
(
Π
1
)
∗
—
cf.
Remark
3.5.1.]
∼
Next,
let
us
observe
that,
for
each
∈
{◦,
•},
since
(
H)
{l}
→
Π
G
H
[{l}]
⊆
(
Π
1
)
∗
is
open,
and
(
Π
1
)
∗
is
topologically
finitely
generated,
slim
[cf.
Proposition
1.7,
(i)]
and
almost
pro-l,
there
exist
an
open
subgroup
J
⊆
I
K
of
I
K
and
a
homomorphism
ρ
1
[
H]
:
J
−→
Out((
H)
{l}
)
such
that
ρ
1
[
H]
is
compatible
[in
the
evident
sense]
with
the
homo-
morphism
J
→
Out((
Π
1
)
∗
)
induced
by
ρ
1
:
I
K
→
Out(
Π
1
),
and,
moreover,
ρ
1
[
H]
factors
through
the
maximal
pro-l
quotient
(
J)
{l}
of
J,
which
[as
is
easily
verified]
is
isomorphic
to
Z
l
as
an
abstract
profinite
group.
Moreover,
it
follows
immediately
from
the
various
defi-
nitions
involved,
together
with
the
well-known
properness
of
the
moduli
stack
of
pointed
stable
curves
of
a
given
type,
that
the
outer
represen-
∼
tation
(
J)
{l}
→
Out((
H)
{l}
)
→
Out(Π
G
H
[{l}]
)
arising
from
such
a
homomorphism
ρ
1
[
H]
is
of
PIPSC-type
[cf.
Definition
1.3].
In
par-
ticular,
it
follows
immediately
from
Theorem
1.11,
(ii)
[i.e.,
in
essence,
[CbTpII],
Theorem
1.9,
(ii)],
that
if
α
satisfies
condition
(b
{l}
),
i.e.,
α
∼
is
{l}-I-admissible,
then
the
isomorphism
of
(
◦
H)
{l}
→
Π
G
◦
H
[{l}]
with
∼
(
•
H)
{l}
→
Π
G
•
H
[{l}]
induced
by
α
is
group-theoretically
verticial,
hence
also
group-theoretically
nodal.
Thus,
by
allowing
“
H”
to
vary
among
the
various
characteristic
open
subgroups
of
Π
1
,
we
conclude
that
if
α
satisfies
condition
(b
{l}
),
i.e.,
α
is
{l}-I-admissible,
then
α
satisfies
condition
(b
∃
)
of
Proposi-
tion
3.6,
hence
[cf.
Proposition
3.6,
(iii);
our
assumption
that
α
is
group-theoretically
cuspidal
if
◦
p
∈
◦
Σ,
•
p
∈
•
Σ]
that
α
is
G-admissible
[cf.
[CmbGC],
Proposition
1.5,
(ii)].
In
particular,
it
follows
from
either
∼
of
the
conditions
(a),
(b
{l}
)
that
the
isomorphism
of
(
◦
H)
{l}
→
Π
G
◦
H
[{l}]
∼
with
(
•
H)
{l}
→
Π
G
•
H
[{l}]
induced
by
α
is
graphic
[cf.
the
implication
COMBINATORIAL
ANABELIAN
TOPICS
III
67
(c
∃
)
⇒
(c
∀
)
of
Proposition
3.6,
(i)],
hence
that
α
determines
a
commu-
tative
diagram
of
isomorphisms
of
profinite
groups
D
G◦
[{l}]
Dehn(G
◦
H
[{l}])
−−−
H
−−→
Node(G
◦
H
[{l}])
Λ
G
◦
H
[{l}]
⏐
⏐
⏐
⏐
−−−−−→
Dehn(G
•
H
[{l}])
D
G•
H
[{l}]
Node(G
•
H
[{l}])
Λ
G
•
H
[{l}]
[cf.
[CbTpI],
Definition
4.4;
[CbTpI],
Theorem
4.8,
(iv)].
On
the
other
hand,
since,
for
each
∈
{◦,
•},
the
outer
repre-
∼
sentation
(
J)
{l}
→
Out((
H)
{l}
)
→
Out(Π
G
H
[{l}]
)
is
of
PIPSC-type,
it
follows
—
by
replacing
J
by
an
open
subgroup
of
J
if
neces-
sary
—
from
[CbTpI],
Corollary
5.9,
(iii),
that
we
may
assume
with-
out
loss
of
generality
that
this
outer
representation
factors
through
Dehn(G
H
[{l}])
⊆
Out(Π
G
H
[{l}]
).
Thus,
by
considering
the
Dehn
co-
ordinates
[cf.
[CbTpI],
Definition
5.8,
(i)]
of
the
image
of
a
topological
generator
of
(
J)
{l}
in
Dehn(G
H
[{l}])
[with
respect
to
a
topological
generator
of
Λ
G
H
[{l}]
],
it
follows
immediately
from
[CbTpI],
Theorem
5.7;
[CbTpI],
Lemma
5.4,
(ii),
together
with
the
existence
of
the
com-
mutative
diagram
of
the
above
display,
that
∼
the
isomorphism
G
◦
H
→
G
•
H
induced
by
α
is
∅-ratio-
nally
compatible
[cf.
Definition
3.4]
with
the
metric
structures
μ
◦
H
,
μ
•
H
[cf.
Definition
3.5,
(iii)]
if
and
only
if
the
images
of
the
homomorphisms
(
◦
J)
{l}
→
Dehn(G
◦
H
[{l}])
and
(
•
J)
{l}
→
Dehn(G
•
H
[{l}])
are
com-
patible,
up
to
a
Q
>0
-multiple,
with
the
isomorphisms
induced
by
α.
In
particular,
by
applying
this
equivalence
to
the
various
characteris-
tic
open
subgroups
“
H”⊆
Π
1
of
Π
1
,
we
conclude
that
α
satisfies
condition
(b
{l}
),
i.e.,
α
is
{l}-I-admissible,
if
and
only
if
α
satisfies
condition
(a),
i.e.,
α
is
M-admissible.
This
completes
the
proof
of
The-
orem
3.9.
Definition
3.10.
In
the
notation
of
Definition
3.1,
let
l
∈
Σ
be
such
that
l
=
p
and
H
⊆
Π
n
an
open
subgroup
of
Π
n
.
For
each
i
∈
{0,
·
·
·
,
n},
write
H
i
⊆
Π
i
for
the
open
subgroup
of
the
quotient
log
Π
n
Π
i
[induced
by
the
projection
(X
K
)
log
n
→
(X
K
)
i
to
the
first
log
i
factors]
determined
by
the
image
of
H
⊆
Π
n
;
Y
i
→
(X
K
)
log
i
for
the
connected
finite
log
étale
covering
of
(X
K
)
log
corresponding
to
i
H
i
⊆
Π
i
.
Then
we
have
a
sequence
of
morphisms
of
log
schemes
log
log
log
log
log
Y
n
−→
Y
n−1
−→
·
·
·
−→
Y
2
−→
Y
1
−→
Y
0
.
68
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
log
Thus,
for
i
∈
{0,
·
·
·
,
n},
if
we
write
U
i
for
the
interior
of
Y
i
[cf.
the
discussion
entitled
“Log
schemes”
in
[CbTpI],
§0],
we
obtain
a
sequence
of
morphisms
of
schemes
[each
of
which
determines
a
family
of
hyperbolic
curves]
U
n
−→
U
n−1
−→
·
·
·
−→
U
2
−→
U
1
−→
U
0
.
Then
we
shall
say
that
H
is
of
l-polystable
type
if
the
following
con-
ditions
are
satisfied:
(a)
For
each
i
∈
{0,
·
·
·
,
n},
α
∈
Aut
F
(
Π
i
)
[cf.
[CmbCsp],
Defi-
nition
1.1,
(ii)],
the
open
subgroup
H
i
⊆
Π
i
is
preserved
by
α.
Here,
for
convenience,
when
n
=
1,
and
Σ
is
arbitrary,
we
def
set
Aut
F
(
Π
1
)
=
Aut(
Π
1
).
[In
particular,
H
i
is
normal.]
(b)
The
[necessarily
F-characteristic
—
cf.
condition
(a)
above;
Definition
2.1,
(iii)]
maximal
almost
pro-l
quotient
∗
(π
1
((X
K
)
log
n
)
)
Π
n
(
Π
n
)
with
respect
to
H
=
H
n
⊆
Π
n
[cf.
Definition
1.1]
is
SA-
maximal
[cf.
Definition
2.1,
(ii)].
(c)
For
each
i
∈
{1,
·
·
·
,
n},
if
we
write
(
H
i/i−1
)
{l}
for
the
maxi-
def
mal
pro-l
quotient
of
the
kernel
H
i/i−1
=
Ker(
H
i
H
i−1
),
then
the
natural
action
of
H
i−1
on
the
l
aut
-abelianization
[cf.
Lemma
2.14]
of
(
H
i/i−1
)
{l}
is
trivial.
Remark
3.10.1.
In
the
notation
of
Definition
3.10:
(i)
Let
us
observe
that
[one
verifies
easily
that]
condition
(c)
of
Definition
3.10
implies
that
the
following
condition
holds:
(d)
For
each
i
∈
{1,
·
·
·
,
n},
the
natural
outer
representation
H
i−1
−→
Out((
H
i/i−1
)
{l}
)
factors
through
a
pro-l
quotient
of
H
i−1
.
Moreover,
it
follows
from
Lemma
2.14,
(ii);
[ExtFam],
Corol-
lary
7.4
[together
with
the
well-known
structure
of
the
sub-
module
of
the
abelianization
of
(
H
i/i−1
)
{l}
generated
by
the
cuspidal
inertia
subgroups
—
cf.
the
proof
of
Lemma
2.14,
(i)],
that
condition
(c)
of
Definition
3.10
also
implies
that
the
fol-
lowing
condition
holds:
(e)
The
sequence
of
morphisms
of
log
schemes
in
Definition
3.10
log
log
log
log
log
Y
n
−→
Y
n−1
−→
·
·
·
−→
Y
2
−→
Y
1
−→
Y
0
COMBINATORIAL
ANABELIAN
TOPICS
III
69
extends
to
the
factorization
log
log
log
log
log
Y
n
−→
Y
n−1
−→
·
·
·
−→
Y
2
−→
Y
1
−→
Y
0
log
associated
to
the
base-change
to
Y
0
of
the
log
polystable
morphism
determined
by
a
[uniquely
determined!]
stable
polycurve
over
the
integral
closure
of
R
in
some
finite
subextension
of
K
in
K
[cf.
[ExtFam],
Definition
4.5].
log
Here,
the
log
structure
of
Y
0
is
the
log
structure
on
Y
0
=
Spec
R
determined
by
the
multiplicative
monoid
of
nonzero
elements
of
R.
(ii)
One
verifies
easily
that,
for
each
i
∈
{0,
·
·
·
,
n},
if
H
⊆
Π
n
is
of
l-polystable
type,
then
H
i
⊆
Π
i
is
of
l-polystable
type.
Next,
we
define
the
notion
of
an
H-l-system.
Roughly
speaking,
the
notion
of
an
H-l-system
may
be
understood
as
a
basis
for
the
topology
of
the
maximal
pro-l-quotient
of
an
open
subgroup
H
⊆
Π
n
consisting
of
open
subgroups
⊆
Π
n
that,
together
with
H
itself,
correspond
to
coverings
that
are
sufficiently
well-behaved
in
various
technical
respects
to
allow
us
to
apply
to
them
the
theory
developed
thus
far
in
the
present
paper.
Definition
3.11.
In
the
notation
of
Definition
3.10,
suppose
that
H
is
of
l-polystable
type
[cf.
Definition
3.10].
(i)
We
shall
write
VCN
sch
(
H)
for
the
set
of
points
y
∈
Y
n
of
the
underlying
scheme
Y
n
of
log
Y
n
[cf.
the
notation
of
condition
(e)
of
Remark
3.10.1,
(i)]
that
satisfy
the
following
condition:
For
i
∈
{0,
·
·
·
,
n},
write
def
log
y
i
∈
Y
i
for
the
image
of
y
in
Y
i
and
y
i
log
=
Y
i
×
Y
i
y
i
.
[Thus,
for
each
i
∈
{1,
·
·
·
,
n},
we
have
a
stable
log
curve
def
log
log
log
log
.]
Then
Y
i
|
y
log
=
Y
i
×
Y
log
y
i−1
over
y
i−1
i−1
i−1
(a)
y
0
is
the
closed
point
of
Y
0
=
Spec
R;
log
(b)
for
each
i
∈
{1,
·
·
·
,
n},
the
point
of
Y
i
|
y
log
determined
i−1
by
y
i
log
is
either
a
cusp,
node,
or
generic
point
[i.e.,
the
generic
point
of
an
irreducible
component]
of
the
stable
log
log
curve
Y
i
|
y
log
.
i−1
Moreover,
we
shall
write
VCN
sch
(
H)
for
the
set
of
elements
y
∈
VCN
sch
(
H)
such
that,
in
the
above
notation,
70
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(c)
for
each
i
∈
{1,
·
·
·
,
n},
the
residue
field
k(y
i−1
)
of
y
i−1
is
separably
closed
in
the
residue
field
k(y
i
)
of
y
i
.
[Here,
we
note
that
this
separably
closedness
assumption
means
that
the
element
∈
VCN
sch
(
H)
under
consideration
corre-
sponds
to
a
single
element
—
i.e.,
as
opposed
to
a
Galois
orbit
of
elements!
—
of
the
set
of
cusps/nodes/vertices
in
the
semi-
graph
determined
by
the
stable
log
curve
fiber
considered
in
(b).]
Thus,
for
each
i
∈
{1,
·
·
·
,
n},
we
have
natural
maps
VCN
sch
(
H)
VCN
sch
(
H
i
),
VCN
sch
(
H)
→
VCN
sch
(
H
i
),
the
first
of
which
is
surjective.
Finally,
we
shall
say
that
H
is
VCN-complete
if
the
equality
VCN
sch
(
H)
=
VCN
sch
(
H)
holds.
(ii)
We
shall
refer
to
a
projective
system
H
=
{
H
λ
}
λ∈Λ
of
open
subgroups
of
Π
n
as
an
H-l-system
if
each
H
λ
is
of
l-polystable
type,
VCN-complete,
and
contained
in
H
[i.e.,
H
λ
⊆
H],
and,
moreover,
Ker
H
(
H)
{l}
=
Ker
Π
n
(
Π
n
)
∗
=
H
λ
λ∈Λ
[cf.
condition
(b)
of
Definition
3.10],
i.e.,
the
system
H
arises
from
a
basis
of
the
topology
of
(
H)
{l}
.
(iii)
Let
H
=
{
H
λ
}
λ∈Λ
be
an
H-l-system
[cf.
(ii)].
Then
we
shall
write
VCN
sch
(
H)
=
lim
VCN
sch
(
H
λ
)
←−
def
λ∈Λ
[cf.
(i)
above;
the
portion
of
[ExtFam],
Corollary
7.4,
concern-
ing
extensions
of
morphisms;
our
assumption
that
each
H
λ
arises
from
an
open
subgroup
of
(
H)
{l}
,
where
l
=
p].
In
fact,
we
shall
see
below
that
VCN
sch
(
H)
is
independent
of
the
choice
of
H
[cf.
Lemma
3.14,
(iv)].
Here,
we
note
that
one
verifies
easily
that,
for
each
i
∈
{0,
·
·
·
,
n},
if
H
=
{
H
λ
}
λ∈Λ
is
an
H-l-system,
and
we
write
(
H
λ
)
i
⊆
Π
i
for
the
image
of
def
H
λ
in
Π
i
,
then
the
system
H
i
=
{(
H
λ
)
i
}
λ∈Λ
is
an
H
i
-
l-system
[cf.
(i),
(ii)
above;
condition
(b)
of
Definition
3.10;
Remark
3.10.1,
(ii)].
Thus,
for
each
i
∈
{0,
·
·
·
,
n},
we
have
a
natural
map
VCN
sch
(
H)
−→
VCN
sch
(
H
i
)
.
Definition
3.12.
In
the
notation
of
Definition
3.11,
let
H
=
{
H
λ
}
λ∈Λ
be
an
H-l-system
[cf.
Definition
3.11,
(ii)]
and
y
∈
VCN
sch
(
H)
[cf.
COMBINATORIAL
ANABELIAN
TOPICS
III
71
Definition
3.11,
(iii)].
For
each
i
∈
{0,
·
·
·
,
n},
write
y
i
∈
VCN
sch
(
H
i
)
for
the
image
of
y
via
the
natural
map
of
the
final
display
of
Defini-
tion
3.11,
(iii).
Let
i
∈
{1,
·
·
·
,
n}.
(i)
Write
G
i,
y
i−1
for
the
semi-graph
of
anabelioids
of
pro-l
PSC-type
determined
by
the
stable
log
curve
constituted
by
the
log
geometric
fiber
log
log
log
of
Y
i
→
Y
i−1
[cf.
Definition
3.11,
(i)]
at
the
point
of
Y
i−1
determined
by
y
i−1
;
G
i,
y
−→
G
i,
y
i−1
i−1
for
the
universal
covering
[corresponding
to
the
[pro-l]
funda-
mental
group
Π
G
i,
yi−1
of
G
i,
y
i−1
relative
to
the
basepoint
of
G
i,
y
i−1
determined
by
the
various
H
λ
’s]
obtained
by
considering
the
“G
i,
y
i−1
’s”
arising
from
the
various
H
λ
’s.
(ii)
Write
VCN
sch
(
H
i
)|
y
i−1
=
{
y
∈
VCN
sch
(
H
i
)
|
y
i−1
=
y
i−1
}
def
[cf.
Definition
3.11,
(iii)].
Then
one
verifies
easily
from
the
various
definitions
involved
that
we
have
a
natural
bijection
∼
VCN
sch
(
H
i
)|
y
−→
VCN(
G
i,
y
)
i−1
i−1
[cf.
(i)].
In
particular,
the
element
y
i
∈
VCN
sch
(
H
i
)|
y
i−1
determines
an
element
z
i,
y
∈
VCN(
G
i,
y
)
i−1
of
VCN(
G
i,
y
i−1
).
(iii)
It
follows
immediately
from
the
various
definitions
involved
that
we
have
a
natural
action
of
(
H
i
)
{l}
,
hence
also
of
(
H
i/i−1
)
{l}
[cf.
the
notation
of
condition
(c)
of
Definition
3.10],
on
the
set
VCN
sch
(
H
i
).
Thus,
we
obtain
a
tautological
isomorphism
∼
Π
G
i,
yi−1
−→
(
H
i/i−1
)
{l}
such
that
the
various
VCN-subgroups
[cf.
[CbTpI],
Definition
2.1,
(i)]
on
the
left-hand
side
of
this
isomorphism
correspond
to
the
various
stabilizer
subgroups
of
(
H
i/i−1
)
{l}
associated
to
elements
of
VCN
sch
(
H
i
)|
y
i−1
[cf.
the
notation
of
(ii);
the
natural
bijection
of
the
second
display
of
(ii)]
on
the
right-hand
side
of
this
isomorphism.
(iv)
Let
(F
i
)
i∈{1,···
,n}
be
a
collection
of
closed
subgroups
F
i
⊆
(
H
i
)
{l}
.
Then
we
shall
say
that
the
collection
(F
i
)
i∈{1,···
,n}
is
the
VCN-
chain
of
H
associated
to
y
∈
VCN
sch
(
H)
if,
for
each
i
∈
{1,
·
·
·
,
n},
the
closed
subgroup
F
i
coincides
with
the
image
of
72
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
the
VCN-subgroup
of
Π
G
i,
yi−1
associated
to
z
i,
y
∈
VCN(
G
i,
y
i−1
)
∼
[cf.
(ii)]
via
the
isomorphism
Π
G
i,
yi−1
→
(
H
i/i−1
)
{l}
⊆
(
H
i
)
{l}
of
(iii).
We
shall
say
that
the
collection
(F
i
)
i∈{1,···
,n}
is
an
H-VCN-chain
of
H
if
(F
i
)
i∈{1,···
,n}
is
the
VCN-chain
of
H
associated
to
an
element
of
VCN
sch
(
H).
Write
VCN
gp
(
H)
for
the
set
of
H-VCN-chains
of
H.
In
fact,
we
shall
see
be-
low
that
VCN
gp
(
H)
is
independent
of
the
choice
of
H
[cf.
Lemma
3.14,
(iv)].
Thus,
we
conclude
from
[CmbGC],
Propo-
sition
1.2,
(i),
that
the
natural
bijections
of
(ii)
determine
a
bijection
∼
VCN
sch
(
H)
−→
VCN
gp
(
H)
.
Definition
3.13.
In
the
notation
of
Definition
3.1:
∼
(i)
We
shall
say
that
an
isomorphism
of
profinite
groups
◦
Π
n
→
•
Π
n
is
SAF-admissible
[i.e.,
“standard-adjacent-fiber-admissi-
ble”]
if
it
is
PF-admissible
whenever
n
≥
2
[cf.
[CbTpI],
Defi-
nition
1.4,
(i)]
and,
moreover,
is
compatible
with
the
standard
fiber
filtrations
on
◦
Π
n
and
•
Π
n
[cf.
[CmbCsp],
Definition
1.1,
∼
(i)].
We
shall
refer
to
an
outer
isomorphism
◦
Π
n
→
•
Π
n
as
SAF-admissible
if
it
arises
from
an
SAF-admissible
isomor-
phism.
One
verifies
easily
that,
in
the
case
of
an
automor-
phism
or
outomorphism,
SAF-admissibility
is
equivalent
to
F-
admissibility
whenever
n
≥
2.
∼
(ii)
Let
α
:
◦
Π
n
→
•
Π
n
be
an
isomorphism
of
profinite
groups
[so
◦
Σ
=
•
Σ
—
cf.,
e.g.,
the
proof
of
[CbTpI],
Proposition
1.5,
(i)]
and
l
∈
◦
Σ
=
•
Σ
such
that
l
∈
{
◦
p,
•
p}.
Then
we
shall
say
that
α
is
{l}-G-admissible
[i.e.,
{l}-graph-admissible]
if
α
is
SAF-
admissible
[cf.
(i)],
and,
moreover,
the
following
condition
is
satisfied:
Let
◦
J
⊆
◦
Π
n
be
an
open
subgroup
of
◦
Π
n
.
Then
there
exist
an
open
subgroup
◦
H
⊆
◦
Π
n
of
◦
Π
n
of
l-polystable
type
[cf.
Definition
3.10]
and
an
◦
H-l-
system
◦
H
=
{
◦
H
λ
}
λ∈Λ
[cf.
Definition
3.11,
(ii)]
such
def
that
◦
H
⊆
◦
J,
•
H
=
α(
◦
H)
is
of
l-polystable
type,
def
•
H
=
{
•
H
λ
=
α(
◦
H
λ
)}
λ∈Λ
is
an
•
H-l-system,
and,
∼
moreover,
the
isomorphism
◦
H
→
•
H
determined
by
α
induces
a
bijection
∼
VCN
gp
(
◦
H)
−→
VCN
gp
(
•
H)
[cf.
Definition
3.12,
(iv)].
COMBINATORIAL
ANABELIAN
TOPICS
III
73
∼
We
shall
say
that
an
outer
isomorphism
◦
Π
n
→
•
Π
n
is
{l}-G-
admissible
if
it
arises
from
an
{l}-G-admissible
isomorphism.
∼
(iii)
We
shall
say
that
an
isomorphism
◦
Π
n
→
•
Π
n
[so
◦
Σ
=
•
Σ
—
cf.,
e.g.,
the
proof
of
[CbTpI],
Proposition
1.5,
(i)]
is
G-
admissible
[i.e.,
graph-admissible]
if
◦
Σ
=
•
Σ
⊆
{
◦
p,
•
p},
and,
moreover,
the
isomorphism
is
{l}-G-admissible
[cf.
(ii)]
for
every
prime
number
l
∈
◦
Σ
=
•
Σ
such
that
l
∈
{
◦
p,
•
p}.
We
∼
shall
say
that
an
outer
isomorphism
◦
Π
n
→
•
Π
n
is
G-admissible
if
it
arises
from
a
G-admissible
isomorphism.
(iv)
We
shall
write
Aut
{l}-G
(
◦
Π
n
)
⊆
Aut(
◦
Π
n
)
for
the
subgroup
[cf.
Lemma
3.14,
(ii),
(iii),
below]
of
{l}-G-
admissible
automorphisms
of
◦
Π
n
[cf.
(ii)];
Out
{l}-G
(
◦
Π
n
)
=
Aut
{l}-G
(
◦
Π
n
)/Inn(
◦
Π
n
)
⊆
Out(
◦
Π
n
)
def
for
the
subgroup
of
{l}-G-admissible
outomorphisms
of
◦
Π
n
;
def
Aut
G
(
◦
Π
n
)
=
Aut
{l}-G
(
◦
Π
n
)
⊆
Aut(
◦
Π
n
)
l∈
◦
Σ\(
◦
Σ∩{
◦
p})
for
the
subgroup
of
G-admissible
automorphisms
of
◦
Π
n
[cf.
(iii)];
def
Out
G
(
◦
Π
n
)
=
Out
{l}-G
(
◦
Π
n
)
⊆
Out(
◦
Π
n
)
l∈
◦
Σ\(
◦
Σ∩{
◦
p})
for
the
subgroup
of
G-admissible
outomorphisms
of
◦
Π
n
.
Remark
3.13.1.
(i)
In
the
notation
of
Definition
3.13,
suppose
that
n
=
1.
Then
it
follows
immediately
from
Proposition
3.6,
(ii);
Lemma
3.14,
(ii),
(iii),
below;
[CmbGC],
Proposition
1.5,
(ii),
that
the
fol-
lowing
conditions
are
equivalent:
•
α
is
G-admissible
in
the
sense
of
Definition
3.7,
(i).
•
There
exists
a
prime
number
l
∈
◦
Σ
=
•
Σ
such
that
l
∈
{
◦
p,
•
p},
and,
moreover,
α
is
{l}-G-admissible
in
the
sense
of
Definition
3.13,
(ii).
•
α
is
G-admissible
in
the
sense
of
Definition
3.13,
(iii).
In
particular,
for
any
prime
number
l
∈
◦
Σ
such
that
l
=
◦
p,
we
have
equalities
Out(
◦
Π
1
)
G
=
Out
G
(
◦
Π
1
)
=
Out
{l}-G
(
◦
Π
1
)
[cf.
Definitions
3.7,
(i);
3.13,
(iv)].
74
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(ii)
In
the
notation
of
Definition
3.13,
(iv),
one
verifies
easily
from
the
various
definitions
involved
that
Out
G
(
◦
Π
n
)
⊆
Out
{l}-G
(
◦
Π
n
)
⊆
Out
FC
(
◦
Π
n
)
[cf.
[CmbCsp],
Definition
1.1,
(ii)].
Lemma
3.14
(Subgroups
of
l-polystable
type).
In
the
notation
of
∼
Definition
3.1,
let
α
:
◦
Π
n
→
•
Π
n
be
an
isomorphism
of
profinite
groups
[so
◦
Σ
=
•
Σ
—
cf.,
e.g.,
the
proof
of
[CbTpI],
Proposition
1.5,
(i)]
and
l
∈
◦
Σ
=
•
Σ
such
that
l
∈
{
◦
p,
•
p}.
Suppose
that
α
is
SAF-admissible
[cf.
Definition
3.13,
(i)].
Then
the
following
hold:
(i)
Let
◦
J
⊆
◦
Π
n
be
an
open
subgroup
of
◦
Π
n
.
Then
there
exists
an
open
subgroup
◦
H
⊆
◦
Π
n
of
◦
Π
n
of
l-polystable
type
[cf.
Definition
3.10]
such
that
◦
H
⊆
◦
J.
(ii)
Let
◦
H
⊆
◦
Π
n
be
an
open
subgroup
of
◦
Π
n
of
l-polystable
def
type.
Then
•
H
=
α(
◦
H)
is
an
open
subgroup
of
•
Π
n
of
l-
polystable
type.
(iii)
Let
◦
H
⊆
◦
Π
n
be
an
open
subgroup
of
l-polystable
type
of
◦
Π
n
.
Then
there
exists
an
◦
H-l-system
◦
H
=
{
◦
H
λ
}
λ∈Λ
[cf.
def
Definition
3.11,
(ii)]
such
that
•
H
=
{
•
H
λ
=
α(
◦
H
λ
)}
λ∈Λ
is
an
•
H-l-system
[cf.
(ii)].
(iv)
Let
◦
H
⊆
◦
Π
n
be
an
open
subgroup
of
l-polystable
type
of
◦
Π
n
;
◦
H
=
{
◦
H
λ
}
λ∈Λ
,
◦
H
†
=
{
◦
H
λ
†
†
}
λ
†
∈Λ
†
◦
H-l-systems.
Then
there
exists
an
◦
H-l-system
◦
H
‡
=
{
◦
H
λ
‡
‡
}
λ
‡
∈Λ
‡
that
satisfies
the
condition
that,
for
each
(λ,
λ
†
)
∈
Λ
×
Λ
†
,
there
ex-
ists
a
λ
‡
∈
Λ
‡
such
that
◦
H
λ
‡
‡
⊆
◦
H
λ
∩
◦
H
λ
†
†
.
In
particular,
the
sets
VCN
sch
(
H)
[cf.
Definition
3.11,
(iii)]
and
VCN
gp
(
H)
[cf.
Definition
3.12,
(iv)]
are
independent
of
the
choice
of
H
[cf.
Definition
3.11,
(ii)],
i.e.,
depend
only
on
◦
H,
re-
spectively.
(v)
Let
◦
H,
◦
H
†
⊆
◦
Π
n
be
open
subgroups
of
l-polystable
type
of
◦
Π
n
;
◦
H
=
{
◦
H
λ
}
λ∈Λ
an
◦
H-l-system;
◦
H
†
=
{
◦
H
λ
†
†
}
λ
†
∈Λ
†
an
◦
H
†
-l-system.
Suppose
that
the
inclusion
◦
H
†
⊆
◦
H,
hence
def
def
also
the
inclusion
•
H
†
=
α(
◦
H
†
)
⊆
•
H
=
α(
◦
H),
holds.
def
Suppose,
moreover,
that
•
H
=
{
•
H
λ
=
α(
◦
H
λ
)}
λ∈Λ
is
an
•
H-
def
l-system
[cf.
(ii)],
and
that
•
H
†
=
{
•
H
λ
†
†
=
α(
◦
H
†
λ
†
)}
λ
†
∈Λ
†
is
∼
an
•
H
†
-l-system
[cf.
(ii)].
Then
if
the
isomorphism
◦
H
†
→
•
†
H
determined
by
α
induces
a
bijection
∼
VCN
gp
(
◦
H
†
)
−→
VCN
gp
(
•
H
†
),
COMBINATORIAL
ANABELIAN
TOPICS
III
75
∼
then
the
isomorphism
◦
H
→
•
H
determined
by
α
induces
a
bijection
∼
VCN
gp
(
◦
H)
−→
VCN
gp
(
•
H).
Proof.
First,
we
verify
assertion
(i)
by
induction
on
n.
Write
◦
J
n−1
for
the
image
of
◦
J
in
◦
Π
n−1
and
(
◦
J
n/n−1
)
{l}
for
the
maximal
pro-l
def
quotient
of
the
kernel
◦
J
n/n−1
=
Ker(
◦
J
◦
J
n−1
).
Now
let
us
observe
that
if
n
=
1,
then
assertion
(i)
follows
immediately
from
the
various
definitions
involved
[cf.
also
the
fact
that
◦
Π
n
is
topologically
finitely
generated
—
cf.
[MzTa],
Proposition
2.2,
(ii)]].
Thus,
suppose
that
n
≥
2,
and
that
the
induction
hypothesis
is
in
force.
Next,
let
us
observe
that
since
◦
Π
n
is
topologically
finitely
generated
[cf.
[MzTa],
Proposition
2.2,
(ii)],
we
may
assume
without
loss
of
gen-
erality
—
by
replacing
◦
J
by
a
suitable
open
subgroup
of
◦
J
—
that
◦
J
satisfies
condition
(a)
of
Definition
3.10
in
the
case
where
we
take
“i”
to
be
n.
Next,
by
applying
the
induction
hypothesis
to
◦
J
n−1
,
we
obtain
an
open
subgroup
◦
H
n−1
⊆
◦
Π
n−1
of
◦
Π
n−1
that
is
contained
in
def
◦
J
n−1
and
of
l-polystable
type.
Write
◦
H
=
◦
H
n−1
×
◦
J
n−1
◦
J.
Thus,
we
have
an
exact
sequence
of
profinite
groups
1
−→
◦
J
n/n−1
−→
◦
H
−→
◦
H
n−1
−→
1.
Then
it
follows
immediately
from
the
condition
imposed
above
on
◦
J,
together
with
the
induction
hypothesis,
that
[by
taking
◦
J
n−1
to
be
sufficiently
small]
we
may
assume
without
loss
of
generality
that
◦
H
satisfies
conditions
(a)
and
(c)
of
Definition
3.10
[hence
also
(d)
of
Remark
3.10.1,
(i)].
On
the
other
hand,
by
considering
the
quotient
◦
out
H
(
◦
J
n/n−1
)
{l}
(
◦
H
n−1
)
{l}
[i.e.,
that
arises
from
the
fact
that
◦
H
satisfies
condition
(d)
of
Remark
3.10.1,
(i)
—
cf.
also
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0],
we
conclude
that
the
natural
homomorphism
(
◦
J
n/n−1
)
{l}
→
(
◦
H)
{l}
induced
by
the
natural
inclusion
◦
J
n/n−1
→
◦
H
is
injective.
Thus,
one
verifies
easily
from
Lemma
1.2,
(i)
[where
we
take
“(G,
N,
J)”
to
be
(
◦
Π
n
,
◦
H,
◦
Π
n−1
)],
(ii)
[where
we
take
“(G,
N,
H)”
to
be
(
◦
Π
n
,
◦
H,
◦
Π
n/n−1
)],
together
with
our
choice
of
◦
H
n−1
,
that
◦
H
satisfies
condition
(b)
of
Definition
3.10,
i.e.,
that
◦
H
is
l-polystable
type.
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
the
various
definitions
involved.
Next,
we
verify
assertions
(iii),
(iv).
Let
us
first
observe
that,
to
verify
assertions
(iii),
(iv),
it
suffices
to
verify
the
following
assertion:
Claim
3.14.A:
Let
◦
J
⊆
◦
H
be
an
open
subgroup
that
arises
from
an
open
subgroup
of
the
maximal
pro-l
quotient
(
◦
H)
{l}
of
◦
H.
Then
there
exists
an
open
def
subgroup
◦
N
⊆
◦
J
such
that
◦
N
,
•
N
=
α(
◦
N
)
are
of
l-polystable
type,
VCN-complete,
and,
moreover,
arise
from
open
subgroups
of
(
◦
H)
{l}
,
(
•
H)
{l}
,
respectively.
76
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
In
the
remainder
of
the
proofs
of
assertions
(iii),
(iv),
we
verify
Claim
def
3.14.A
by
induction
on
n.
Write
•
J
=
α(
◦
J).
For
each
∈
{◦,
•},
write
J
n−1
for
the
image
of
J
in
Π
n−1
and
(
J
n/n−1
)
{l}
for
the
max-
def
imal
pro-l
quotient
of
the
kernel
J
n/n−1
=
Ker(
J
J
n−1
).
Now
let
us
observe
that
if
n
=
1,
then
Claim
3.14.A
follows
immediately
from
the
various
definitions
involved
[cf.
also
the
fact
that
◦
Π
n
is
topo-
logically
finitely
generated
—
cf.
[MzTa],
Proposition
2.2,
(ii)]].
Thus,
suppose
that
n
≥
2,
and
that
the
induction
hypothesis
is
in
force.
Next,
let
us
observe
that
since
◦
Π
n
is
topologically
finitely
generated
[cf.
[MzTa],
Proposition
2.2,
(ii)],
and
◦
H
satisfies
condition
(a)
of
Def-
inition
3.10,
we
may
assume
without
loss
of
generality
—
by
replacing
◦
J
by
a
suitable
open
subgroup
of
◦
J
—
that
◦
J
satisfies
condition
(a)
of
Definition
3.10
in
the
case
where
we
take
“i”
to
be
n.
Next,
let
us
observe
that
since
◦
J
arises
from
an
open
subgroup
of
(
◦
H)
{l}
,
by
con-
∼
out
sidering
the
natural
isomorphism
(
◦
H)
{l}
→
(
◦
H
n/n−1
)
{l}
(
◦
H
n−1
)
{l}
[i.e.,
that
arises
from
the
fact
that
◦
H
satisfies
condition
(d)
of
Re-
mark
3.10.1,
(i)],
we
conclude
that
◦
J
satisfies
condition
(d)
of
Re-
mark
3.10.1,
(i),
in
the
case
where
we
take
“i”
to
be
n.
In
particular,
since
the
natural
action
of
◦
J
n−1
on
((
◦
J
n/n−1
)
{l}
)
ab
⊗
Z
Z/l
aut
Z
factors
through
a
pro-l
quotient
of
◦
J
n−1
,
we
may
assume
without
loss
of
gen-
erality
—
by
replacing
◦
J
by
the
inverse
image
in
◦
J
of
a
suitable
open
subgroup
of
◦
J
n−1
—
that
◦
J
satisfies
condition
(c)
of
Definition
3.10
in
the
case
where
we
take
“i”
to
be
n.
Thus,
by
applying
the
induction
hypothesis
to
◦
J
n−1
⊆
◦
H
n−1
,
we
def
obtain
an
open
subgroup
◦
N
n−1
⊆
◦
J
n−1
such
that
◦
N
n−1
,
•
N
n−1
=
α(
◦
N
n−1
)
are
of
l-polystable
type,
VCN-complete,
and
arise
from
open
subgroups
of
(
◦
H
n−1
)
{l}
,
(
•
H
n−1
)
{l}
,
respectively.
Write
◦
N
=
◦
N
n−1
×
◦
J
n−1
◦
J.
def
Then
one
verifies
immediately,
by
a
similar
argument
to
the
argument
def
applied
in
the
final
portion
of
the
proof
of
assertion
(i),
that
◦
N
,
•
N
=
α(
◦
N
)
are
of
l-polystable
type
and,
moreover,
arise
from
open
subgroups
of
(
◦
H)
{l}
,
(
•
H)
{l}
,
respectively.
In
particular,
since
◦
N
,
•
N
satisfy
condition
(d)
of
Remark
3.10.1,
(i),
in
the
case
where
we
take
“i”
to
be
n,
we
may
assume
without
loss
of
generality
—
by
replacing
◦
N
by
the
inverse
image
in
◦
N
of
a
suitable
open
subgroup
of
◦
N
n−1
[cf.
the
induction
hypothesis]
—
that
◦
N
,
•
N
satisfy
the
condition
that
each
of
the
elements
of
VCN
sch
(
◦
N
),
VCN
sch
(
•
N
)
satisfies
condition
(c)
of
Definition
3.11,
(i),
in
the
case
where
we
take
“i”
to
be
n.
Thus,
we
conclude
[cf.
the
fact
that
◦
N
n−1
,
•
N
n−1
are
VCN-complete]
that
◦
N
,
•
N
are
VCN-complete.
This
completes
the
proof
of
Claim
3.14.A,
hence
also
the
proofs
of
assertions
(iii),
(iv).
COMBINATORIAL
ANABELIAN
TOPICS
III
77
Finally,
we
verify
assertion
(v).
For
each
∈
{◦,
•}
and
each
†
i
∈
{1,
.
.
.
,
n},
write
H
i/i−1
,
H
i/i−1
for
the
respective
subquotients
†
of
H,
H
determined
by
the
subquotient
Π
i/i−1
of
Π
n
;
(
H
i/i−1
)
{l}
,
†
†
(
H
i/i−1
)
{l}
for
the
respective
maximal
pro-l
quotients
of
H
i/i−1
,
H
i/i−1
[cf.
Definition
3.10,
(c)].
Then
let
us
observe
that
it
follows
immedi-
ately
from
[CmbGC],
Proposition
1.2,
(ii),
together
with
the
various
definitions
involved
that,
for
each
∈
{◦,
•}
and
each
i
∈
{1,
.
.
.
,
n},
every
VCN-subgroup
of
(
H
i/i−1
)
{l}
[i.e.,
discussed
as
in
Definition
3.12,
(iii),
(iv)]
may
be
obtained
as
the
commensurator
of
the
image
of
a
†
†
VCN-subgroup
of
(
H
i/i−1
)
{l}
[by
the
homomorphism
(
H
i/i−1
)
{l}
→
†
(
H
i/i−1
)
{l}
determined
by
the
natural
inclusion
H
⊆
H].
More-
over,
one
also
verifies
easily
that
every
proper
closed
subgroup
of
(
H
i/i−1
)
{l}
obtained
as
the
commensurator
of
the
image
of
a
VCN-subgroup
of
†
(
H
i/i−1
)
{l}
is
a
VCN-subgroup
of
(
H
i/i−1
)
{l}
.
Assertion
(v)
now
fol-
lows
formally.
This
completes
the
proof
of
Lemma
3.14.
Definition
3.15.
In
the
notation
of
Definition
3.12,
write
(F
i
)
i∈{1,···
,n}
∈
VCN
gp
(
H)
for
the
VCN-chain
of
H
associated
to
y
∈
VCN
sch
(
H)
[cf.
Definition
3.12,
(iv)].
Now
since
(
H)
{l}
⊆
(
Π
n
)
∗
[cf.
the
notation
of
condition
(b)
of
Definition
3.10]
is
open,
and
(
Π
n
)
∗
is
topologically
finitely
generated,
slim
[cf.
Proposition
2.3,
(i)]
and
almost
pro-l,
there
exist
an
open
subgroup
J
⊆
I
K
of
I
K
and
a
homomorphism
ρ
:
J
−→
Out((
H)
{l}
)
that
•
is
compatible
[in
the
evident
sense]
with
the
homomorphism
J
→
Out((
Π
n
)
∗
)
induced
[cf.
condition
(a)
of
Definition
3.10]
by
ρ
n
:
I
K
→
Out(
Π
n
),
•
induces,
for
each
i
∈
{1,
·
·
·
,
n},
a
homomorphism
J
−→
Out((
H
i
)
{l}
)
—
relative
to
the
natural
surjection
(
H)
{l}
(
H
i
)
{l}
—
and,
moreover,
•
factors
through
the
maximal
pro-l
quotient
(
J)
{l}
of
J,
which
[as
is
easily
verified]
is
isomorphic
to
Z
l
as
an
abstract
profinite
group.
def
Write
I
y
0
=
(
J)
{l}
.
Then,
for
i
∈
{1,
·
·
·
,
n},
we
define
closed
sub-
groups
I
y
i
⊆
ρ
H
i
|
y
i−1
⊆
ρ
def
out
H
i
=
(
H
i
)
{l}
(
J)
{l}
78
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0]
as
fol-
lows
[inductively
on
i]:
(i)
Set
ρ
ρ
H
1
|
y
0
=
H
1
,
I
y
1
=
Z
H
ρ
1
|
y
(F
1
).
def
def
0
(ii)
Suppose
that
n
≥
i
≥
2.
Then,
by
the
induction
hypothesis,
we
have
already
constructed
closed
subgroups
I
y
i−1
⊆
ρ
H
i−1
|
y
i−2
⊆
ρ
H
i−1
,
hence
also
a
natural
outer
representation
ρ
I
y
i−1
→
H
i−1
→
Out((
H
i/i−1
)
{l}
)
—
where
the
second
arrow
is
the
natural
outer
representation
arising
from
the
exact
sequence
of
profinite
groups
ρ
ρ
1
−→
(
H
i/i−1
)
{l}
−→
H
i
−→
H
i−1
−→
1.
Then
we
set
ρ
out
H
i
|
y
i−1
=
(
H
i/i−1
)
{l}
I
y
i−1
,
I
y
i
=
Z
H
ρi
|
y
def
def
i−1
(F
i
).
Remark
3.15.1.
In
the
situation
of
Definition
3.15,
it
follows
imme-
diately
from
the
definition
of
I
y
i
[cf.
also
[CmbGC],
Remark
1.1.3;
[CmbGC],
Proposition
1.2,
(ii)]
that
I
y
i
is
isomorphic
to
a
profinite
group
of
the
form
Z
⊕j
l
,
where
j
is
a
positive
integer
≤
i
+
1.
Proposition
3.16
(Graph-admissible
isomorphisms).
In
the
no-
∼
tation
of
Definition
3.1,
let
α
:
◦
Π
n
→
•
Π
n
be
an
isomorphism
of
profi-
nite
groups
[so
◦
Σ
=
•
Σ
—
cf.,
e.g.,
the
proof
of
[CbTpI],
Proposition
1.5,
(i)]
and
l
∈
◦
Σ
=
•
Σ
such
that
l
∈
{
◦
p,
•
p}.
Then
the
following
hold:
(i)
If
◦
p
∈
◦
Σ
and
•
p
∈
•
Σ,
then
suppose
that
α
is
PC-admissible
[cf.
[CbTpI],
Definition
1.4,
(ii)].
If
α
is
SAF-admissible
[cf.
Definition
3.13,
(i)]
and
{l}-I-admissible
[cf.
Defini-
tion
3.8,
(i)],
then
α
is
{l}-G-admissible
[cf.
Definition
3.13,
(ii)].
(ii)
Suppose
that
α
is
{l}-G-admissible.
Then
there
exists
an
algorithm,
which
is
functorial
with
respect
to
α,
for
con-
structing
an
isomorphism
of
topological
groups
∼
•
tp
α
tp
:
◦
Π
tp
n
−→
Π
n
∼
such
that
the
isomorphism
◦
Π
n
→
•
Π
n
induced
by
α
tp
[cf.
Proposition
3.3,
(i)]
coincides
with
α.
COMBINATORIAL
ANABELIAN
TOPICS
III
79
Proof.
First,
we
verify
assertion
(i).
Let
◦
J
⊆
◦
Π
n
be
an
open
sub-
group
of
◦
Π
n
.
Then
it
follows
from
Lemma
3.14,
(i),
(ii),
(iii),
that
there
exist
an
open
subgroup
◦
H
⊆
◦
Π
n
of
◦
Π
n
of
l-polystable
type
[cf.
Definition
3.10]
and
an
◦
H-l-system
◦
H
=
{
◦
H
λ
}
λ∈Λ
[cf.
Defini-
def
tion
3.11,
(ii)]
such
that
◦
H
⊆
◦
J,
•
H
=
α(
◦
H)
is
of
l-polystable
type,
def
and
•
H
=
{
•
H
λ
=
α(
◦
H
λ
)}
λ∈Λ
is
an
•
H-l-system.
Now
it
follows
im-
mediately
from
the
various
definitions
involved
that,
to
complete
the
verification
of
assertion
(i),
it
suffices
to
verify
the
following
assertion:
Claim
3.16.A:
For
each
i
∈
{1,
·
·
·
,
n},
the
isomor-
∼
phism
◦
H
i
→
•
H
i
[cf.
the
notation
of
Definition
3.10]
determined
by
α
induces
a
bijection
∼
VCN
gp
(
◦
H
i
)
−→
VCN
gp
(
•
H
i
)
[cf.
Definitions
3.11,
(iii);
3.12,
(iv)].
We
verify
Claim
3.16.A
by
induction
on
i.
If
i
=
1,
then
Claim
3.16.A
follows
immediately
from
the
equivalence
(a)
⇔
(b
∃
)
of
Theorem
3.9,
together
with
Remark
3.13.1,
(i).
Now
suppose
that
i
≥
2,
and
that
the
induction
hypothesis
is
in
force.
Then
it
follows
immediately
from
the
induction
hypothesis
that,
for
each
j
∈
{1,
·
·
·
,
i
−
1},
the
isomorphism
∼
◦
H
j
→
•
H
j
determined
by
α
induces
a
bijection
∼
VCN
gp
(
◦
H
j
)
−→
VCN
gp
(
•
H
j
).
Let
◦
y
i−1
∈
VCN
sch
(
◦
H
i−1
),
•
y
i−1
∈
VCN
sch
(
•
H
i−1
)
[cf.
Definition
3.11,
(iii)]
be
elements
that
correspond
via
the
above
bijection,
relative
to
the
◦-,
•-versions
of
the
displayed
bijection
of
Definition
3.12,
(iv).
Now
since
α
is
{l}-I-admissible,
for
∈
{◦,
•},
there
exist
an
open
subgroup
J
⊆
I
K
of
I
K
and
an
outer
representation
ρ
:
J
→
Out((
H)
{l}
)
as
in
Definition
3.15
such
that
◦
ρ
is
compatible,
relative
to
α,
with
•
ρ.
Thus,
it
follows
immediately
from
the
various
definitions
∼
involved
that
the
isomorphism
◦
H
i
→
•
H
i
determined
by
α
induces
an
isomorphism
of
profinite
groups
◦
∼
H
ρi
|
◦
y
i−1
−→
•
H
ρi
|
•
y
i−1
∼
that
lies
over
an
isomorphism
β
:
I
◦
y
i−1
→
I
•
y
i−1
[cf.
Definition
3.15].
In
particular,
we
obtain
a
commutative
diagram
of
profinite
groups
I
◦
y
i−1
−−−→
Out((
◦
H
i/i−1
)
{l}
)
⏐
⏐
⏐
⏐
β
I
•
y
i−1
−−−→
Out((
•
H
i/i−1
)
{l}
)
—
where
the
right-hand
vertical
arrow
is
the
isomorphism
induced
by
α.
Moreover,
one
verifies
immediately
from
the
various
definitions
involved
[cf.
also
Remark
3.15.1]
that,
for
each
∈
{◦,
•},
the
pos-
itive
definite
profinite
Dehn
multi-twists
[cf.
[CbTpI],
Definition
4.4;
80
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[CbTpI],
Definition
5.8,
(iii)]
in
the
image
of
the
composite
∼
I
y
i−1
−→
Out((
H
i/i−1
)
{l}
)
←−
Out(Π
G
i,
y
i−1
)
—
where
the
second
arrow
is
the
isomorphism
induced
by
the
isomor-
phism
of
Definition
3.12,
(iii)
—
form
a
dense
subset
of
this
image
[cf.
[CbTpI],
Lemma
5.4,
(i),
(ii),
(iii);
[CbTpI],
Proposition
5.6,
(ii)].
In
particular,
it
follows
immediately
[cf.
the
easily
verified
elementary
fact
that
no
dense
subset
of
a
nonzero
finitely
generated
free
Z
l
-module
is
contained
in
a
finite
union
of
proper
Z
l
-submodules
of
the
given
finitely
generated
free
Z
l
-module]
that
there
exists
an
element
◦
γ
∈
I
◦
y
i−1
such
def
that
if
we
write
•
γ
=
β(
◦
γ)
∈
I
•
y
i−1
,
then,
for
=
◦
(respectively,
=
•),
the
image
of
γ
via
the
composite
of
the
above
display
is
a
positive
definite
profinite
Dehn
multi-twist
(respectively,
nondegenerate
profinite
Dehn
multi-twist
[cf.
[CbTpI],
Definition
4.4;
[CbTpI],
Defi-
nition
5.8,
(ii)]).
Thus,
it
follows
immediately
from
[CbTpII],
Theorem
1.9,
(ii),
together
with
the
equivalences
of
[CbTpI],
Corollary
5.9,
(ii),
(iii),
that
the
isomorphism
∼
∼
∼
α
i/i−1
:
Π
G
i,
◦
y
i−1
−→
(
◦
H
i/i−1
)
{l}
−→
(
•
H
i/i−1
)
{l}
←−
Π
G
i,
•
y
i−1
induced
by
α
is
group-theoretically
verticial,
hence
also
group-theoretically
nodal.
Next,
let
us
observe
that
it
follows
from
the
fact
that
α
i/i−1
is
group-
theoretically
verticial
[hence
also
group-theoretically
nodal],
together
with
our
assumption
concerning
PC-admissibility,
that
if
◦
p
∈
◦
Σ
and
•
p
∈
•
Σ,
then
[cf.
[CmbGC],
Proposition
1.5,
(ii)]
α
i/i−1
is
graphic.
On
the
other
hand,
if
either
◦
p
∈
◦
Σ
or
•
p
∈
•
Σ,
then
it
follows
from
Propo-
sition
3.6,
(iii)
[applied
to
“(c
∃
)”
—
cf.
Remark
3.13.1,
(i)],
together
with
Claim
3.16.A
in
the
case
where
i
=
1,
that
◦
p
=
•
p
∈
◦
Σ
=
•
Σ.
In
particular,
if
either
◦
p
∈
◦
Σ
or
•
p
∈
•
Σ,
then,
by
allowing
the
open
sub-
group
“
◦
H”
of
◦
Π
n
to
vary
and
applying
the
group-theoretic
nodality
of
the
resulting
isomorphisms
“α
i/i−1
”,
one
concludes
from
the
“existence
of
irreducible
components
that
collapse
to
arbitrary
cusps”
[cf.
the
proof
of
“observation
(iv)”
given
in
the
proof
of
[SemiAn],
Corollary
3.11;
[SemiAn],
Remark
3.11.1;
[AbsTpII],
Corollary
2.11;
[AbsTpII],
Remark
2.11.1,
(i)]
that
α
i/i−1
is
group-theoretically
cuspidal,
hence
also
[cf.
[CmbGC],
Proposition
1.5,
(ii)]
graphic.
Thus,
by
allowing
◦
y
i−1
,
•
y
i−1
to
vary,
we
conclude
immediately
from
the
various
definitions
involved
that
Claim
3.16.A
holds.
This
completes
the
proof
of
Claim
3.16.A,
hence
also
of
assertion
(i).
Next,
we
verify
assertion
(ii).
The
theory
of
[Brk]
yields
•
a
functorial
homotopy
[indeed,
a
proper
strong
deformation
retraction!]
between
the
skeleton
of
a
polystable
fibration
[cf.
COMBINATORIAL
ANABELIAN
TOPICS
III
81
[Brk],
Definitions
1.2,
1.3]
over
the
ring
of
integers
of
a
com-
plete
nonarchimedean
field
and
the
analytic
space
associated
to
the
polystable
fibration
[cf.
[Brk],
Theorem
8.1],
as
well
as
•
a
functorial
homeomorphism
between
the
skeleton
of
a
polystable
fibration
over
the
ring
of
integers
of
a
complete
nonarchimedean
field
and
the
geometric
realization
of
a
certain
polysimplicial
set
associated
to
the
special
fiber
of
the
polystable
fibration
[cf.
[Brk],
Theorem
8.2].
In
particular,
the
theory
of
[Brk]
gives
rise
to
a
functorial
homotopy
between
the
analytic
space
associated
to
a
polystable
fibration
over
the
ring
of
integers
of
a
complete
nonar-
chimedean
field
and
the
geometric
realization
of
a
cer-
tain
polysimplicial
set
associated
to
the
special
fiber
of
the
polystable
fibration.
Here,
we
recall
further
that
this
polysimplicial
set
is
completely
de-
termined
by
the
set
of
strata
of
the
special
fiber,
together
with
the
specialization/generization
relations
between
these
strata
[cf.
the
dis-
cussion
surrounding
[Brk],
Proposition
2.1,
and
its
proof;
[Brk],
Lemma
3.13;
[Brk],
Lemma
6.7].
Next,
let
us
observe
that
the
various
bijections
∼
∼
∼
VCN
sch
(
◦
H)
−→
VCN
gp
(
◦
H)
−→
VCN
gp
(
•
H)
←−
VCN
sch
(
•
H)
[cf.
Definitions
3.12,
(iv);
3.13,
(ii)]
induced
by
an
{l}-G-admissible
iso-
∼
morphism
◦
Π
n
→
•
Π
n
induce
bijections
between
the
respective
sets
of
strata
of
the
special
fibers
of
◦
Y
n
,
•
Y
n
[cf.
the
notation
of
condition
(e)
of
Remark
3.10.1,
(i)],
which,
in
light
of
the
group-theoretic
descriptions
of
specialization/generization
relations
given
in
[CbTpI],
Proposition
2.9,
(i)
[cf.
also
[CbTpI],
Proposition
5.6,
(iii),
(iv)],
are
[easily
seen
to
be]
compatible
with
these
specialization/generization
relations.
In
par-
log
ticular,
since
each
log
scheme
Y
n
gives
rise
to
a
polystable
fibration
as
in
the
above
discussion
of
[Brk]
[cf.
condition
(e)
of
Remark
3.10.1,
(i)],
we
thus
conclude,
in
light
of
the
theory
of
[Brk],
from
the
defini-
tion
of
the
tempered
fundamental
group
given
in
[André],
§4.2
[cf.
also
the
discussion
of
Definition
3.1,
(ii),
of
the
present
paper],
that
any
∼
{l}-G-admissible
isomorphism
◦
Π
n
→
•
Π
n
determines
an
isomorphism
◦
∼
•
tp
Π
tp
n
−→
Π
n
between
the
respective
tempered
fundamental
groups,
which
gives
back
∼
the
original
isomorphism
◦
Π
n
→
•
Π
n
upon
passing
to
the
respective
◦
Σ
=
•
Σ-completions
[cf.
Proposition
3.3,
(i)].
This
completes
the
proof
of
assertion
(ii).
82
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Theorem
3.17
(Metric-,
inertia-admissible
outomorphisms
of
fundamental
groups).
Let
n
be
a
positive
integer;
(g,
r)
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
p
a
prime
number;
Σ
a
nonempty
set
of
prime
numbers
such
that
Σ
=
{p},
and,
moreover,
if
n
≥
2,
then
Σ
is
either
equal
to
the
set
of
all
prime
numbers
or
of
cardinality
one;
R
a
mixed
characteristic
complete
discrete
valuation
ring
of
residue
characteristic
p
whose
residue
field
is
separably
closed;
K
the
field
of
fractions
of
R;
K
an
algebraic
closure
of
K;
log
X
K
a
smooth
log
curve
of
type
(g,
r)
over
K.
Write
(X
K
)
log
n
for
the
n-th
log
configuration
space
[cf.
the
discussion
entitled
def
log
log
over
K;
(X
K
)
log
“Curves”
in
[CbTpII],
§0]
of
X
K
n
=
(X
K
)
n
×
K
K;
def
Σ
Π
n
=
π
1
((X
K
)
log
n
)
for
the
maximal
pro-Σ
quotient
of
the
log
fundamental
group
of
(X
K
)
log
n
;
def
ρ
n
:
I
K
=
Gal(K/K)
−→
Out(Π
n
)
log
for
the
natural
outer
pro-Σ
Galois
action
associated
to
(X
K
)
log
n
;
(Spec
R)
for
the
log
scheme
obtained
by
equipping
Spec
R
with
the
log
structure
determined
by
the
closed
point
of
Spec
R.
Then
the
following
hold:
(i)
Let
l
∈
Σ
be
such
that
l
=
p.
Then
we
have
equalities
and
an
inclusion
Out(Π
1
)
M
=
Out
I
(Π
1
)
∩
Out
C
(Π
1
)
=
Out
{l}-I
(Π
1
)
∩
Out
C
(Π
1
)
⊆
Out(Π
1
)
G
[cf.
Definitions
3.7,
(i),
(ii);
3.8,
(iii)].
If,
moreover,
p
∈
Σ,
then
we
have
equalities
and
inclusions
Out(Π
1
)
M
=
Out
I
(Π
1
)
=
Out
{l}-I
(Π
1
)
⊆
Out(Π
1
)
G
⊆
Out(Π
1
).
(ii)
Let
l
∈
Σ
be
such
that
l
=
p.
Then
we
have
equalities
and
inclusions
Out
FC
(Π
n
)
M
=
Out
FCI
(Π
n
)
=
Out
FC
(Π
n
)
I
=
Out
FC
{l}-I
(Π
n
)
=
Out
FC
(Π
n
)
{l}-I
⊆
Out
G
(Π
n
)
⊆
Out
{l}-G
(Π
n
),
Out
FC
(Π
n
)
M
⊆
Out
FI
(Π
n
)
⊆
Out
F
{l}-I
(Π
n
)
∩
∩
∩
F
F
F
M
I
⊆
Out
(Π
n
)
⊆
Out
(Π
n
)
{l}-I
Out
(Π
n
)
[cf.
Definitions
3.7,
(iii);
3.8,
(iii),
(iv);
3.13,
(iv)].
Moreover,
the
following
hold:
COMBINATORIAL
ANABELIAN
TOPICS
III
83
(ii-a)
If
p
∈
Σ,
then
we
have:
Out
FC
(Π
n
)
M
=
Out
FI
(Π
n
)
=
Out
F
{l}-I
(Π
n
),
Out
F
(Π
n
)
M
=
Out
F
(Π
n
)
I
=
Out
F
(Π
n
)
{l}-I
.
(ii-b)
If
n
=
1,
then
we
have:
Out
FI
(Π
n
)
=
Out
F
{l}-I
(Π
n
),
Out
F
(Π
n
)
M
=
Out
F
(Π
n
)
I
=
Out
F
(Π
n
)
{l}-I
.
(ii-c)
If
n
=
2,
(r,
n)
=
(0,
3),
and
either
p
∈
Σ
or
n
=
1,
then
we
have:
Out
F
(Π
n
)
M
=
Out
FI
(Π
n
)
=
Out
F
(Π
n
)
I
=
Out
F
{l}-I
(Π
n
)
=
Out
F
(Π
n
)
{l}-I
FC
=
Out
(Π
n
)
M
=
Out
FCI
(Π
n
)
=
Out
FC
(Π
n
)
I
=
Out
FC
{l}-I
(Π
n
)
=
Out
FC
(Π
n
)
{l}-I
.
log
(iii)
Suppose
that
p
∈
Σ,
and
that
X
K
extends
to
a
stable
log
log
curve
over
(Spec
R)
.
Let
l
∈
Σ.
Write
ρ
n
(I
K
)[l]
⊆
ρ
n
(I
K
)
for
the
maximal
pro-l
subgroup
of
the
[necessarily
pro-cyclic
—
cf.
the
injectivity
portion
of
[NodNon],
Theorem
B;
the
dis-
cussion
of
[CbTpI],
Definition
5.3]
image
ρ
n
(I
K
).
Then
the
normalizers
of
ρ
n
(I
K
),
ρ
n
(I
K
)[l]
in
Out
F
(Π
n
)
satisfy
the
fol-
lowing
equalities:
(iii-a)
If
(r,
n)
=
(0,
2),
then
Out
FI
(Π
n
)
=
Out
F
(Π
n
)
I
=
N
Out
F
(Π
n
)
(ρ
n
(I
K
)),
Out
F
{l}-I
(Π
n
)
=
Out
F
(Π
n
)
{l}-I
=
N
Out
F
(Π
n
)
(ρ
n
(I
K
)[l]).
(iii-b)
For
arbitrary
r
≥
0,
n
≥
1,
Out
FI
(Π
n
)
=
N
Out
F
(Π
n
)
(ρ
n
(I
K
)),
Out
F
{l}-I
(Π
n
)
=
N
Out
F
(Π
n
)
(ρ
n
(I
K
)[l]).
(iv)
Let
l
∈
Σ
be
such
that
l
=
p.
Then
the
subgroups
Out(Π
1
)
M
,
Out
I
(Π
1
),
Out
{l}-I
(Π
1
),
Out(Π
1
)
G
of
Out(Π
1
)
are
closed
in
Out(Π
1
).
Moreover,
the
subgroups
Out
F
(Π
n
)
M
,
Out
FI
(Π
n
),
Out
F
(Π
n
)
I
,
F
{l}-I
(Π
n
),
Out
F
(Π
n
)
{l}-I
,
Out
Out
FC
(Π
n
)
M
,
Out
FCI
(Π
n
),
Out
FC
(Π
n
)
I
,
Out
FC
{l}-I
(Π
n
),
Out
FC
(Π
n
)
{l}-I
,
Out
{l}-G
(Π
n
)
Out
G
(Π
n
),
of
Out(Π
n
)
are
closed
in
Out(Π
n
).
In
particular,
these
sub-
groups
are
compact.
84
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(v)
Let
l
∈
Σ
be
such
that
l
=
p.
Suppose,
moreover,
that
the
log
arises,
via
base-change,
from
a
smooth
smooth
log
curve
X
K
log
curve
over
a
complete
discrete
valuation
field
whose
residue
field
is
finitely
generated
over
a
finite
field.
Then
the
closed
subgroups
Out
G
(Π
n
),
Out
{l}-G
(Π
n
)
⊆
Out
F
(Π
n
)
[cf.
(iv);
Remark
3.13.1,
(ii)]
are
commensurably
terminal
in
Out
F
(Π
n
).
Moreover,
we
have
an
inclusion
C
Out
F
(Π
n
)
(Out
FC
(Π
n
)
M
)
⊆
Out
G
(Π
n
).
(vi)
The
natural
homomorphism
Out
FC
(Π
n+1
)
M
−→
Out
FC
(Π
n
)
M
(respectively,
Out
F
(Π
n+1
)
M
−→
Out
F
(Π
n
)
M
)
log
induced
by
the
projection
(X
K
)
log
n+1
→
(X
K
)
n
obtained
by
for-
getting
any
one
of
the
n
+
1
factors
is
injective
(respectively,
injective
if
(r,
n)
=
(0,
1)).
If,
moreover,
either
n
≥
4
or
n
≥
3
and
r
=
0,
then
this
homomorphism
is
bijective
(respectively,
bijective).
Proof.
Assertion
(i)
follows
immediately
from
Theorem
3.9.
Next,
we
verify
assertion
(ii).
First,
we
claim
that
the
following
assertion
holds:
Claim
3.17.A:
We
have
equalities
Out
FC
(Π
n
)
I
=
Out
FCI
(Π
n
);
Out
FC
(Π
n
)
{l}-I
=
Out
FC
{l}-I
(Π
n
).
Indeed,
this
follows
immediately
—
in
light
of
the
definition
of
I-
admissibility,
{l}-I-admissibility
[cf.
Definition
3.8]
—
from
Proposi-
tion
2.3,
(ii),
and
Corollary
2.10
[when
Σ
=
Primes];
the
injectivity
portion
of
[NodNon],
Theorem
B
[when
Σ
=
{l}].
This
completes
the
proof
of
Claim
3.17.A.
Next,
we
claim
that
the
following
assertion
holds:
Claim
3.17.B:
We
have
equalities
Out
FC
(Π
n
)
M
=
Out
FC
(Π
n
)
I
=
Out
FC
(Π
n
)
{l}-I
.
Indeed,
this
follows
immediately
from
assertion
(i),
together
with
the
various
definitions
involved.
This
completes
the
proof
of
Claim
3.17.B.
Next,
we
claim
that
the
following
assertion
holds:
Claim
3.17.C:
We
have
equalities
and
an
inclusion
Out
FC
(Π
n
)
M
=
Out
FC
(Π
n
)
I
=
Out
FC
(Π
n
)
{l}-I
=
Out
FCI
(Π
n
)
=
Out
FC
{l}-I
(Π
n
)
⊆
Out
G
(Π
n
).
COMBINATORIAL
ANABELIAN
TOPICS
III
85
Indeed,
the
first
four
equalities
follow
from
Claims
3.17.A,
3.17.B.
On
the
other
hand,
the
final
inclusion
follows
immediately
from
Proposi-
tion
3.16,
(i)
[cf.
also
the
final
portion
of
Definition
3.13,
(i)].
This
completes
the
proof
of
Claim
3.17.C.
Next,
we
claim
that
the
following
assertion
holds:
Claim
3.17.D:
We
have
inclusions
Out
FC
(Π
n
)
M
⊆
Out
FI
(Π
n
)
⊆
Out
F
{l}-I
(Π
n
)
∩
∩
∩
F
F
F
M
I
Out
(Π
n
)
⊆
Out
(Π
n
)
⊆
Out
(Π
n
)
{l}-I
.
Indeed,
let
us
observe
that
the
left-hand
upper
inclusion
follows
imme-
diately
from
Claim
3.17.C.
Next,
let
us
observe
that
the
left-hand
lower
inclusion
follows
immediately
from
assertion
(i).
On
the
other
hand,
the
remaining
inclusions
follow
immediately
from
the
various
defini-
tions
involved.
This
completes
the
proof
of
Claim
3.17.D.
The
various
equalities
and
inclusions
of
assertion
(ii)
that
precede
assertion
(ii-a)
all
follow
from
Claims
3.17.C,
3.17.D.
Next,
we
consider
assertion
(ii-a).
It
follows
immediately
from
Propo-
sition
3.16,
(i),
that
the
inclusion
Out
F
{l}-I
(Π
n
)
⊆
Out
{l}-G
(Π
n
)
holds.
In
particular,
it
follows
from
Remark
3.13.1,
(ii),
that
the
inclusion
Out
F
{l}-I
(Π
n
)
⊆
Out
FC
(Π
n
),
hence
also
the
equality
Out
F
{l}-I
(Π
n
)
=
Out
FC
{l}-I
(Π
n
),
holds.
Thus,
the
first
two
equalities
of
assertion
(ii-a)
follow
immediately
from
Claims
3.17.C,
3.17.D.
On
the
other
hand,
the
final
two
equalities
of
assertion
(ii-a)
follow
immediately
from
the
final
portion
of
assertion
(i).
This
completes
the
proof
of
assertion
(ii-a).
Next,
we
consider
assertion
(ii-b).
If
p
∈
Σ,
then
assertion
(ii-b)
follows
from
assertion
(ii-a).
Thus,
we
may
assume
without
loss
of
generality
that
p
∈
Σ.
Then
since
[by
assumption!]
Σ
=
{l},
the
first
equality
of
assertion
(ii-b)
follows
immediately
from
the
various
definitions
involved.
On
the
other
hand,
the
final
two
equalities
follow
immediately
from
assertion
(i),
together
with
[CbTpI],
Theorem
A,
(ii).
This
completes
the
proof
of
assertion
(ii-b).
Next,
we
consider
assertion
(ii-c).
If
n
≥
3
and
(r,
n)
=
(0,
3),
then
assertion
(ii-c)
follows
immediately
from
[CbTpII],
Theorem
A,
(ii),
together
with
Claim
3.17.C.
On
the
other
hand,
if
p
∈
Σ
and
n
=
1,
then
assertion
(ii-c)
follows
immediately
from
the
final
portion
of
assertion
(i),
together
with
Claim
3.17.C
[cf.
also
Remark
3.13.1,
(ii)].
This
completes
the
proof
of
assertion
(ii-c),
hence
also
of
assertion
(ii).
Next,
we
verify
assertion
(iii).
First,
we
claim
that
the
following
assertion
holds:
Claim
3.17.E:
We
have
an
equality
Out
{l}-I
(Π
1
)
=
N
Out(Π
1
)
(ρ
1
(I
K
)[l]).
Indeed,
let
us
first
observe
that
since
p
∈
Σ,
we
have
a
natural
outer
∼
isomorphism
Π
1
→
Π
G
Π1
[Σ]
[cf.
Remark
3.5.1].
Next,
let
us
observe
86
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
log
that,
in
light
of
our
assumption
that
X
K
extends
to
a
stable
log
curve
over
(Spec
R)
log
,
it
follows
from
[CbTpI],
Definition
5.3,
(i),
that
the
image
of
ρ
1
is
contained
in
∼
Dehn(G
Π
1
[Σ])
⊆
Out(Π
G
Π1
[Σ]
)
←
Out(Π
1
)
[cf.
[CbTpI],
Definition
4.4]
and,
moreover,
is
pro-cyclic.
Next,
let
us
observe
that
it
follows
immediately
from
the
definition
of
{l}-I-
admissibility
that
N
Out(Π
1
)
(ρ
1
(I
K
)[l])
⊆
Out
{l}-I
(Π
1
).
Thus,
to
com-
plete
the
verification
of
Claim
3.17.E,
it
suffices
to
verify
that
we
have
an
inclusion
Out
{l}-I
(Π
1
)
⊆
N
Out(Π
1
)
(ρ
1
(I
K
)[l]).
Let
α
∈
Out
{l}-I
(Π
1
)
and
H
⊆
Π
1
a
characteristic
open
subgroup
of
Π
1
.
Write
Π
1
Π
∗
1
for
the
maximal
almost
pro-l
quotient
of
Π
1
with
respect
to
H
[cf.
Definition
1.1];
Π
G
Π1
[Σ]
Π
∗G
Π
[Σ]
for
the
[necessarily
1
maximal
almost
pro-l]
quotient
of
Π
G
Π1
[Σ]
corresponding
to
Π
1
Π
∗
1
∼
[relative
to
the
above
natural
outer
isomorphism
Π
1
→
Π
G
Π1
[Σ]
].
Then
it
follows
immediately
[in
light
of
the
well-known
simple
structure
of
pro-cyclic
profinite
groups]
from
the
definition
of
{l}-I-admissibility
that
there
exists
an
open
subgroup
J
⊆
I
K
such
that
the
image
of
ρ
1
(J)
in
Out(Π
∗
1
)
is
normalized
by
the
outomorphism
α
∗
∈
Out(Π
∗
1
)
determined
by
α
∈
Out(Π
1
).
On
the
other
hand,
it
follows
immediately
from
the
above
discussion
that
the
outer
representation
∼
ρ
1
(J)
−→
Out(Π
∗
1
)
−→
Out(Π
∗G
Π
[Σ]
)
1
is
of
PIPSC-type
[cf.
Definition
1.6,
(iv)].
Thus,
it
follows
from
The-
orem
1.11,
(ii),
that
α
∗
∈
Out(Π
∗
1
)
is
group-theoretical
verticial
[cf.
Definition
1.6,
(ii)].
In
particular,
by
allowing
H
to
vary,
we
con-
clude
that
α
∈
Out(Π
1
)
is
group-theoretical
verticial.
Thus,
it
follows
immediately
from
the
definition
of
a
profinite
Dehn
multi-twist
that
∼
α
∈
Out(Π
1
)
normalizes
Dehn(G
Π
1
[Σ])
⊆
Out(Π
G
Π1
[Σ]
)
←
Out(Π
1
),
hence
also
[cf.
[CbTpI],
Theorem
4.8,
(iv)]
the
maximal
pro-l
subgroup
Dehn(G
Π
1
[Σ])[l]
of
Dehn(G
Π
1
[Σ]).
On
the
other
hand,
one
verifies
im-
mediately
again
from
[CbTpI],
Theorem
4.8,
(iv),
that
Dehn(G
Π
1
[Σ])[l]
is
a
free
Z
l
-module
of
finite
rank,
and
that
the
composite
Dehn(G
Π
1
[Σ])[l]
→
Out(Π
1
)
→
Out(Π
∗
1
)
is
injective.
Thus,
since
some
open
subgroup
of
the
maximal
pro-l
sub-
group
of
the
image
of
I
K
in
Out(Π
∗
1
)
is
normalized
by
α
∗
∈
Out(Π
∗
1
)
[cf.
the
above
discussion
concerning
“J”!],
one
verifies
immediately
[from
well-known
elementary
properties
of
free
Z
l
-modules
of
finite
rank]
that
α
∈
N
Out(Π
1
)
(ρ
1
(I
K
)[l]).
This
completes
the
proof
of
Claim
3.17.E.
Now
let
us
observe
that
one
verifies
easily
[cf.
also
the
discussion
of
the
inclusion
“N
Out(Π
1
)
(ρ
1
(I
K
)[l])
⊆
Out
{l}-I
(Π
1
)”
in
the
proof
of
Claim
3.17.E]
that
the
inclusions
N
Out
F
(Π
n
)
(ρ
n
(I
K
))
⊆
Out
FI
(Π
n
)
⊆
Out
F
(Π
n
)
I
,
COMBINATORIAL
ANABELIAN
TOPICS
III
87
N
Out
F
(Π
n
)
(ρ
n
(I
K
)[l])
⊆
Out
F
{l}-I
(Π
n
)
⊆
Out
F
(Π
n
)
{l}-I
hold.
In
particular,
assertion
(iii-a)
follows
immediately
from
Claim
3.17.E,
together
with
the
injectivity
portion
of
[CbTpII],
Theorem
A,
(i)
[cf.
also
[CbTpI],
Theorem
A,
(ii);
[NodNon],
Theorem
B,
in
the
case
where
r
=
0].
Thus,
to
complete
the
proof
of
assertion
(iii),
it
suffices
to
verify
the
two
equalities
of
assertion
(iii-b)
in
the
case
where
(r,
n)
=
(0,
2).
Suppose
that
(r,
n)
=
(0,
2),
hence
that
Σ
=
{l}.
Then
one
verifies
easily
that,
to
complete
the
proof
of
assertion
(iii),
it
suffices
to
verify
that
Out
F
{l}-I
(Π
n
)
⊆
N
Out
F
(Π
n
)
(ρ
n
(I
K
)[l]).
Thus,
let
α
∈
Aut(Π
n
)
be
a
lifting
of
an
element
α
∈
Out
F
{l}-I
(Π
n
).
Then
let
us
observe
that
it
follows
immediately
from
Claim
3.17.E
that
out
α
induces
an
automorphism
β
of
the
extension
group
Π
1
ρ
1
(I
K
)[l]
[i.e.,
arising
from
the
outer
representation
of
IPSC-type
ρ
1
(I
K
)
→
Out(Π
1
)
implicit
in
the
discussion
surrounding
Claim
3.17.E
above],
whose
restriction
to
Π
1
is
G-admissible
[cf.
assertion
(i)].
In
particular,
out
it
follows
that
β
maps
verticial
inertia
groups
of
Π
1
ρ
1
(I
K
)[l]
[each
of
which
surjects
onto
ρ
1
(I
K
)[l]
—
cf.
[NodNon],
Definition
2.2,
(i);
[NodNon],
Definition
2.4,
(ii);
[NodNon],
Remark
2.4.2]
to
verticial
in-
out
ertia
groups
of
Π
1
ρ
1
(I
K
)[l].
Moreover,
let
us
observe
that
it
follows
immediately
from
the
fact
that
α
∈
Out
F
{l}-I
(Π
n
)
that
β
is
compat-
ible
with
the
natural
outer
representations
of
suitable
open
subgroups
out
of
such
verticial
inertia
groups
of
Π
1
ρ
1
(I
K
)[l]
on
Π
2/1
[relative
to
the
action
of
α
on
Π
2/1
].
Thus,
since
the
natural
outer
representation
out
of
such
a
verticial
inertia
group
of
Π
1
ρ
1
(I
K
)[l]
on
Π
2/1
is
[eas-
ily
verified
to
be]
an
outer
representation
of
IPSC-type,
one
concludes
from
a
similar
argument
to
the
[final
portion
of
the]
argument
applied
above
to
verify
Claim
3.17.E
that
β
is
compatible
with
these
natural
out
outer
representations
of
verticial
inertia
groups
of
Π
1
ρ
1
(I
K
)[l]
on
on
Π
2/1
].
Now
it
follows
formally
Π
2/1
[relative
to
the
action
of
α
that
α
∈
N
Out
F
(Π
n
)
(ρ
n
(I
K
)[l]),
as
desired.
This
completes
the
proof
of
assertion
of
(iii).
Next,
we
verify
assertion
(iv).
The
closedness
of
Out(Π
1
)
G
in
Out(Π
1
)
follows
immediately
from
condition
(c
∀
)
of
Proposition
3.6
[cf.
Propo-
sition
3.6,
(ii)].
Thus,
the
closedness
of
Out(Π
1
)
M
in
Out(Π
1
)
follows
from
the
easily
verified
fact
that
Out
M
(Π
1
)
is
closed
in
Out(Π
1
)
G
.
The
fact
that
the
subgroup
Out
{l}-I
(Π
1
),
hence
also
Out
I
(Π
1
),
is
closed
in
Out(Π
1
)
may
be
verified
as
follows:
If
p
∈
Σ,
then
the
closedness
in
question
follows
from
the
closedness
of
Out(Π
1
)
M
[verified
above],
to-
gether
with
the
final
portion
of
assertion
(i).
On
the
other
hand,
if
p
∈
Σ,
then
the
closedness
in
question
follows
immediately
from
as-
sertion
(iii).
This
completes
the
proof
of
the
closedness
of
Out
{l}-I
(Π
1
)
and
Out
I
(Π
1
)
in
Out(Π
1
).
88
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
The
closedness
of
Out
FC
(Π
n
)
M
,
Out
FC
(Π
n
)
I
,
Out
FC
(Π
n
)
{l}-I
,
Out
F
(Π
n
)
M
,
Out
F
(Π
n
)
I
,
Out
F
(Π
n
)
{l}-I
in
Out(Π
n
)
follows
immediately
from
the
various
definitions
involved,
together
with
the
closedness
of
Out(Π
1
)
M
,
Out
I
(Π
1
),
and
Out
{l}-I
(Π
1
)
in
Out(Π
1
)
[verified
above].
The
closedness
of
Out
FCI
(Π
n
),
Out
FC
{l}-I
(Π
n
)
in
Out(Π
n
)
follow
from
the
closedness
of
Out
FC
(Π
n
)
M
in
Out(Π
n
)
[ver-
ified
above],
together
with
the
equalities
at
the
beginning
of
assertion
(ii).
Next,
we
verify
the
closedness
of
Out
G
(Π
n
),
Out
{l}-G
(Π
n
)
in
Out(Π
n
).
Let
us
first
observe
that
it
is
immediate
that,
to
verify
the
desired
closedness,
it
suffices
to
verify
the
closedness
of
Out
{l}-G
(Π
n
)
in
Out(Π
n
).
Next,
to
verify
the
closedness
of
Out
{l}-G
(Π
n
)
in
Out(Π
n
),
let
(α
ξ
)
ξ≥1
be
a
sequence
[indexed
by
the
positive
integers]
of
elements
∈
Out
{l}-G
(Π
n
)
that
converges
to
an
element
α
∞
∈
Out(Π
n
).
Then
since
[one
verifies
easily
that]
the
subgroup
of
Out(Π
n
)
consisting
of
SAF-admissible
outomorphisms
is
closed,
α
∞
is
SAF-admissible.
Next,
to
verify
that
α
∞
satisfies
the
condition
of
Definition
3.13,
(ii),
let
us
fix
an
open
subgroup
J
⊆
Π
n
of
Π
n
.
Now
we
define
open
subgroups
H
ξ
⊆
J
ξ
⊆
Π
n
of
Π
n
inductively
on
ξ
as
follows:
def
•
Set
J
1
=
J.
•
Suppose
that
J
ξ
⊆
Π
n
has
already
been
defined.
Then
since
α
ξ
is
{l}-G-admissible,
there
exists
an
open
subgroup
H
⊆
◦
Π
n
of
◦
Π
n
of
l-polystable
type
such
that
H
⊆
J
ξ
,
and,
moreover,
α
ξ
satisfies
the
condition
of
Definition
3.13,
(ii),
in
the
case
where
we
take
the
“(
◦
J,
◦
H)”
of
Definition
3.13,
(ii),
to
be
(J
ξ
,
H).
def
Then
define
H
ξ
=
H.
•
Suppose
that
ξ
≥
2,
and
that
H
ξ−1
⊆
Π
n
has
already
been
def
defined.
Then
set
J
ξ
=
H
ξ−1
.
Then
it
follows
immediately
from
Lemma
3.14,
(iii),
(v),
that
α
∞
sat-
isfies
the
condition
of
Definition
3.13,
(ii),
in
the
case
where
we
take
the
“(
◦
J,
◦
H)”
of
Definition
3.13,
(ii),
to
be
(J
=
J
1
,
H
=
H
1
).
In
particular,
the
SAF-admissible
outomorphism
α
∞
is
{l}-G-admissible,
as
desired.
This
completes
the
proof
of
the
closedness
of
Out
{l}-G
(Π
n
)
in
Out(Π
n
).
The
fact
that
the
subgroup
Out
F
{l}-I
(Π
n
),
hence
also
Out
FI
(Π
n
),
is
closed
in
Out(Π
n
)
may
be
verified
as
follows:
If
n
=
1,
then
the
closed-
ness
in
question
has
already
been
verified.
If
p
∈
Σ,
then
the
closedness
in
question
follows
from
the
closedness
of
Out
FC
(Π
n
)
M
[verified
above],
COMBINATORIAL
ANABELIAN
TOPICS
III
89
together
with
assertion
(ii-a).
On
the
other
hand,
if
p
∈
Σ,
then
the
closedness
in
question
follows
from
assertion
(iii-b).
This
completes
the
proof
of
the
closedness
of
Out
F
{l}-I
(Π
n
),
Out
FI
(Π
n
)
in
Out(Π
n
),
hence
also
of
assertion
(iv).
Next,
we
verify
assertion
(v).
Let
α
∈
C
Out
F
(Π
n
)
(Out
G
(Π
n
))
(re-
spectively,
C
Out
F
(Π
n
)
(Out
{l}-G
(Π
n
));
C
Out
F
(Π
n
)
(Out
FC
(Π
n
)
M
))
and
α
∈
F
Aut
(Π
n
)
a
lifting
of
α.
Now
observe
that
to
complete
the
verifi-
cation
of
assertion
(v),
it
suffices
to
verify
that
α
∈
Out
{l}-G
(Π
n
).
To
this
end,
let
J
⊆
Π
n
be
an
open
subgroup
of
Π
n
.
Then
it
fol-
lows
from
Lemma
3.14,
(i),
(iii),
that
there
exist
an
open
subgroup
H
⊆
J
⊆
Π
n
of
Π
n
of
l-polystable
type
[cf.
Definition
3.10]
and
an
H-l-system
H
=
{H
λ
}
λ∈Λ
[cf.
Definition
3.11,
(ii)].
Note
that
it
fol-
lows
from
condition
(a)
of
Definition
3.10
that
the
subgroups
H,
H
λ
of
Π
n
are
stabilized
by
α
.
Then
it
follows
immediately
from
the
various
definitions
involved
that,
to
complete
the
verification
of
the
fact
that
α
∈
Out
{l}-G
(Π
n
),
it
suffices
to
verify
the
following
assertion:
Claim
3.17.F:
For
each
i
∈
{0,
·
·
·
,
n},
the
outomor-
phism
of
the
image
H
i
of
H
in
Π
i
determined
by
α
induces
a
bijection
∼
VCN
gp
(H
i
)
−→
VCN
gp
(H
i
)
[cf.
Definition
3.11,
(iii);
Definition
3.12,
(iv)]
—
where,
def
def
for
convenience,
we
set
Π
0
=
{1},
VCN
gp
(H
0
)
=
{Π
0
},
and
we
write
(H
λ
)
i
for
the
image
of
H
λ
in
Π
i
def
and
H
i
=
{(H
λ
)
i
}
λ∈Λ
.
We
verify
Claim
3.17.F
by
induction
on
i.
If
i
=
0,
then
Claim
3.17.F
is
immediate.
Now
suppose
that
i
≥
1,
and
that
the
induction
hypothesis
is
in
force.
Then
it
follows
immediately
from
the
induction
hypothesis
that,
for
each
j
∈
{0,
·
·
·
,
i
−
1},
the
outomorphism
of
H
j
determined
by
α
induces
a
bijection
∼
VCN
gp
(H
j
)
−→
VCN
gp
(H
j
).
Let
y
,
y
∈
VCN
sch
(H
i−1
)
[cf.
Definition
3.11,
(iii)]
be
elements
that
correspond
via
the
bijection
obtained
by
conjugating
the
above
bijec-
tion
by
the
displayed
bijection
of
Definition
3.12,
(iv).
Next,
let
us
observe
that
since
α
∈
C
Out
F
(Π
n
)
(Out
G
(Π
n
))
(respec-
tively,
C
Out
F
(Π
n
)
(Out
{l}-G
(Π
n
));
C
Out
F
(Π
n
)
(Out
FC
(Π
n
)
M
)),
there
exist
open
subgroups
N
1
and
N
2
of
Out
G
(Π
n
)
(respectively,
Out
{l}-G
(Π
n
);
extends
Out
FC
(Π
n
)
M
)
such
that
the
automorphism
of
H
i
induced
by
α
to
an
isomorphism
of
profinite
groups
[cf.
assertion
(iv)]
out
∼
out
H
i
N
1
−→
H
i
N
2
90
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0]
that
lies
∼
over
an
isomorphism
of
profinite
groups
N
1
→
N
2
.
In
particular,
by
out
considering
the
respective
outer
actions
[by
conjugation]
of
H
i−1
N
1
,
out
H
i−1
N
2
on
the
maximal
pro-l
quotient
(H
i/i−1
)
{l}
of
the
kernel
def
H
i/i−1
=
Ker(H
i
H
i−1
)
[cf.
the
notation
of
Remark
3.10.1,
(i)],
we
obtain
a
commutative
diagram
of
profinite
groups
out
∼
out
∼
H
i−1
N
1
−−−→
Out((H
i/i−1
)
{l}
)
←−−−
Out(Π
G
i,
y
)
⏐
⏐
⏐
⏐
⏐
⏐
H
i−1
N
2
−−−→
Out((H
i/i−1
)
{l}
)
←−−−
Out(Π
G
i,
y
)
—
where
the
left-hand
vertical
arrow
is
the
isomorphism
induced
by
the
isomorphism
of
profinite
groups
discussed
above;
the
central
verti-
cal
arrow
is
the
automorphism
induced
by
α
;
the
right-hand
horizontal
arrows
are
the
isomorphisms
induced
by
the
y
-,
y
-versions
of
the
iso-
morphism
of
Definition
3.12,
(iii);
the
right-hand
vertical
arrow
is
the
isomorphism
induced
by
the
composite
∼
∼
∼
α
y
,
y
:
Π
G
i,
y
−→
(H
i/i−1
)
{l}
−→
(H
i/i−1
)
{l}
←−
Π
G
i,
y
∼
of
the
isomorphism
Π
G
i,
y
→
(H
i/i−1
)
{l}
of
Definition
3.12,
(iii),
the
automorphism
of
(H
i/i−1
)
{l}
determined
by
α
,
and
the
isomorphism
{l}
∼
(H
i/i−1
)
←
Π
G
i,
y
of
Definition
3.12,
(iii).
Now
let
us
recall
that
we
have
assumed
that
the
smooth
log
curve
log
X
K
arises,
via
base-change,
from
a
smooth
log
curve
over
a
complete
discrete
valuation
field
whose
residue
field
is
finitely
generated
over
a
finite
field.
In
particular,
one
verifies
immediately
from
the
openness
of
N
1
,
N
2
in
Out
G
(Π
n
)
(respectively,
Out
{l}-G
(Π
n
);
Out
FC
(Π
n
)
M
=
Out
FC
(Π
n
)
I
⊆
Out
{l}-G
(Π
n
)
[cf.
assertion
(ii)])
that
the
composite
horizontal
arrows
of
the
above
commutative
diagram
factor
through
Aut(G
i,
y
),
Aut(G
i,
y
),
respectively,
and,
moreover,
are
l-graphically
full
[i.e.,
in
the
sense
of
[CmbGC],
Definition
2.3,
(iii)]
—
cf.
the
argu-
ment
applied
in
the
proof
of
[CmbGC],
Proposition
2.4,
(v).
Thus,
it
follows
from
[CmbGC],
Corollary
2.7,
(ii),
that
the
isomorphism
∼
α
y
,
y
:
Π
G
i,
y
→
Π
G
i,
y
is
graphic.
In
particular,
by
allowing
y
,
y
to
vary,
it
follows
immediately
from
the
various
definitions
involved
that
Claim
3.17.F
holds.
This
completes
the
proof
of
Claim
3.17.F,
hence
also
of
assertion
(v).
Assertion
(vi)
follows
from
[NodNon],
Theo-
rem
B;
[CbTpII],
Theorem
A,
(i).
This
completes
the
proof
of
Theo-
rem
3.17.
COMBINATORIAL
ANABELIAN
TOPICS
III
91
Remark
3.17.1.
In
the
notation
of
Theorem
3.17,
suppose
that
we
are
in
the
situation
of
Theorem
3.17,
(v).
Then
it
follows
from
Theo-
rem
3.17,
(v),
that
C
Out
F
(Π
n
)
(Out
FC
(Π
n
)
M
)
⊆
Out
G
(Π
n
).
On
the
other
hand,
Out
FC
(Π
n
)
M
is
not,
in
general,
commensurably
terminal
in
Out
G
(Π
n
)
[or
indeed
in
Out
F
(Π
n
)
or
Out
FC
(Π
n
)!].
Indeed,
suppose,
moreover,
that
we
are
in
the
situation
of
Theorem
3.17,
(iii)
[so
p
∈
Σ],
and
that
the
semi-graph
of
anabelioids
G
of
pro-Σ
PSC
type
log
determined
by
the
geometric
special
fiber
of
the
stable
model
of
X
K
satisfies
the
following
conditions:
•
Vert(G)
=
Node(G)
=
2.
Write
Vert(G)
=
{v
1
,
v
2
},
Node(G)
=
{e
1
,
e
2
}.
•
For
each
i
∈
{1,
2},
V(e
i
)
=
Vert(G)
=
{v
1
,
v
2
}.
•
There
exists
an
automorphism
of
G
that
induces
a
nontrivial
automorphism
of
Node(G).
Finally,
suppose
that
if
we
write
μ
X
log
for
the
metric
structure
on
the
K
log
underlying
semi-graph
of
G
associated
to
the
stable
model
of
X
K
[cf.
Definition
3.5,
(iii)],
then
μ
X
log
(e
1
)
=
μ
X
log
(e
2
).
[Here,
we
note
that
one
K
K
log
verifies
easily
that
such
a
smooth
log
curve
X
K
exists.]
Then
it
follows
immediately
from
the
various
assumptions
imposed
on
the
objects
un-
der
consideration
that
Out
FC
(Π
1
)
M
is
of
index
2,
hence
also
normal,
in
Out
G
(Π
1
).
In
particular,
Out
FC
(Π
1
)
M
is
not
normally
terminal,
hence,
a
fortiori,
not
commensurably
terminal,
in
Out
G
(Π
1
).
Remark
3.17.2.
In
the
notation
of
Theorem
3.17,
suppose
that
p
∈
Σ.
(i)
It
follows
from
Theorem
3.17,
(ii-c),
that
if
either
(†
1
):
n
≥
4
or
n
≥
3
and
r
=
0,
then
we
have
equalities
Out
F
(Π
n
)
M
=
Out
FC
(Π
n
)
M
=
Out
FI
(Π
n
)
=
Out
F
{l}-I
(Π
n
)
=
Out
FCI
(Π
n
)
=
Out
FC
{l}-I
(Π
n
)
=
Out
F
(Π
n
)
I
=
Out
F
(Π
n
)
{l}-I
=
Out
FC
(Π
n
)
I
=
Out
FC
(Π
n
)
{l}-I
.
(ii)
In
Corollary
2.10,
the
authors
gave
what
may
be
regarded
as
an
almost
pro-l
version
of
the
injectivity
portion
of
[NodNon],
Theorem
B
[i.e.,
the
injectivity
of
the
natural
homomorphism
Out
FC
(Π
n+1
)
→
Out
FC
(Π
n
)].
In
fact,
however,
although
a
de-
tailed
exposition
lies
beyond
the
scope
of
the
present
paper
[cf.
the
discussion
of
(iii)
below],
it
seems
quite
likely
that
92
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
it
should
be
possible
to
verify
an
almost
pro-l
version
of
the
injectivity
portion
of
[CbTpII],
Theorem
A,
(i)
[i.e.,
the
injec-
tivity
of
the
natural
homomorphism
Out
F
(Π
n+1
)
→
Out
F
(Π
n
)
for
(r,
n)
=
(0,
1)].
Such
an
almost
pro-l
version
would
then
imply,
via
a
similar
argument
to
the
argument
applied
in
the
proof
of
the
equalities
Out
FCI
(Π
n
)
=
Out
FC
(Π
n
)
I
,
Out
FC
{l}-I
(Π
n
)
=
Out
FC
(Π
n
)
{l}-I
[cf.
Claim
3.17.A
in
the
proof
of
Theorem
3.17,
(ii)],
that
if
either
(†
2
):
n
≥
3
or
n
≥
2
and
r
=
0,
then
the
equalities
Out
FI
(Π
n
)
=
Out
F
(Π
n
)
I
,
Out
F
{l}-I
(Π
n
)
=
Out
F
(Π
n
)
{l}-I
,
hence
also
[cf.
Theorem
3.17,
(ii);
Theorem
3.17,
(ii-a)]
the
nine
equalities
of
the
display
of
(i),
hold.
(iii)
The
main
reason
that
the
authors
did
not
go
to
the
trouble
to
verify
the
nine
equalities
of
the
display
of
(i)
under
the
more
general
hypotheses
[i.e.,
(†
2
)]
discussed
in
(ii)
is
the
following.
The
main
applications
of
the
theory
developed
in
the
present
paper
are
the
following:
(1)
the
generalization,
given
in
Corollary
3.20
below
[cf.
also
Remark
3.20.1
below],
of
a
result
due
to
Andre
[cf.
[André],
Theorems
7.2.1,
7.2.3]
concerning
the
characterization
of
local
Galois
groups
in
the
global
Galois
image
associated
to
a
hyperbolic
curve
over
a
number
field
and
(2)
the
establishment
of
an
appropriate
local
analogue,
sat-
isfying
various
expected
properties,
of
the
Grothendieck-
Teichmüller
group
[cf.
Remark
3.19.2
below].
The
theory
surrounding
these
applications
[cf.
Theorem
3.18
below]
revolves
around
the
theory
of
the
tripod
homomorphism
developed
in
[CbTpII],
§3.
On
the
other
hand,
this
theory
of
the
tripod
homomorphism
is
only
well-behaved
[cf.
[CbTpII],
Definition
3.19]
under
the
more
restrictive
hypotheses
[i.e.,
(†
1
)]
discussed
in
(i).
Theorem
3.18
(Metric-admissible
outomorphisms
and
tripods).
In
the
notation
of
Theorem
3.17,
the
following
hold:
(i)
Suppose
that
n
≥
3.
Let
Π
tpd
be
a
1-central
{1,
2,
3}-tripod
of
Π
n
[cf.
[CbTpII],
Definitions
3.3,
(i);
3.7,
(ii)].
Then
the
restriction
of
the
tripod
homomorphism
associated
to
Π
n
T
Π
tpd
:
Out
FC
(Π
n
)
−→
Out
C
(Π
tpd
)
COMBINATORIAL
ANABELIAN
TOPICS
III
93
[cf.
[CbTpII],
Definition
3.19]
to
the
subgroup
Out
FC
(Π
n
)
M
⊆
Out
FC
(Π
n
)
[cf.
Definition
3.7,
(iii)]
factors
through
the
sub-
group
Out(Π
tpd
)
M
⊆
Out
C
(Π
tpd
)
[cf.
Definition
3.7,
(i),
(ii);
Remark
3.13.1,
(i),
(ii)],
i.e.,
we
have
a
natural
commutative
diagram
Out
FC
(Π
n
)
M
−−−→
Out(Π
tpd
)
M
⏐
⏐
⏐
⏐
Out
FC
(Π
n
)
−−−→
Out
C
(Π
tpd
).
T
Πtpd
(ii)
Suppose
that
n
≥
1,
and
that
(g,
r)
=
(0,
3).
Write
Out
F
(Π
n
)
Δ+
⊆
Out
F
(Π
n
)
for
the
inverse
image
via
the
natural
homomorphism
Out
F
(Π
n
)
→
Out(Π
1
)
[cf.
[CbTpI],
Theorem
A,
(i)]
of
Out
C
(Π
1
)
Δ+
⊆
Out(Π
1
)
[cf.
[CbTpII],
Definition
3.4,
(i)];
def
Out
FC
(Π
n
)
Δ+
=
Out
F
(Π
n
)
Δ+
∩
Out
FC
(Π
n
)
[cf.
Remark
3.18.1
below];
def
Out
F
(Π
n
)
MΔ+
=
Out
F
(Π
n
)
Δ+
∩
Out
F
(Π
n
)
M
;
def
Out
FC
(Π
n
)
MΔ+
=
Out
FC
(Π
n
)
Δ+
∩
Out
F
(Π
n
)
M
.
Then
we
have
equalities
Out
F
(Π
n
)
Δ+
=
Out
FC
(Π
n
)
Δ+
,
Out
F
(Π
n
)
MΔ+
=
Out
FC
(Π
n
)
MΔ+
.
Moreover,
the
natural
homomorphisms
Out
FC
(Π
n+1
)
Δ+
−−−→
Out
FC
(Π
n
)
Δ+
Out
F
(Π
n+1
)
Δ+
−−−→
Out
F
(Π
n
)
Δ+
Out
FC
(Π
n+1
)
MΔ+
−−−→
Out
FC
(Π
n
)
MΔ+
Out
F
(Π
n+1
)
MΔ+
−−−→
Out
F
(Π
n
)
MΔ+
are
bijective.
Proof.
Assertion
(i)
follows
immediately
—
in
light
of
the
equalities
Out
FC
(Π
n
)
M
=
Out
FCI
(Π
n
)
,
Out(Π
tpd
)
M
=
Out
I
(Π
tpd
)
∩
Out
C
(Π
tpd
)
94
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[cf.
Theorem
3.17,
(i),
(ii)]
—
from
the
definition
of
I-admissibility,
together
with
[in
the
case
where
Σ
=
Primes]
Corollary
2.13,
(iii).
Next,
we
verify
assertion
(ii).
The
equalities
Out
F
(Π
n
)
Δ+
=
Out
FC
(Π
n
)
Δ+
,
Out
F
(Π
n
)
MΔ+
=
Out
FC
(Π
n
)
MΔ+
follow
immediately
from
[CbTpII],
Theorem
A,
(ii),
together
with
the
various
definitions
involved.
Next,
let
us
observe
that,
to
verify
the
bijectivity
of
the
various
homomorphisms
in
question,
it
suffices
to
verify
the
bijectivity
of
the
natural
homomorphism
Out
FC
(Π
n+1
)
Δ+
−−−→
Out
FC
(Π
n
)
Δ+
.
On
the
other
hand,
this
bijectivity
follows
immediately,
in
light
of
the
various
definitions
involved,
from
[CmbCsp],
Corollary
4.2,
(i),
(ii).
This
completes
the
proof
of
assertion
(ii),
hence
also
of
Theorem
3.18.
Remark
3.18.1.
In
the
notation
of
Theorem
3.18,
suppose
that
n
≥
2.
Then
in
[CmbCsp],
Definition
1.11,
(ii),
a
definition
was
given
for
the
notation
“Out
FC
(Π
n
)
Δ+
”,
in
the
case
of
arbitrary
(g,
r),
that
differs
somewhat
from
the
definition
given
for
this
notation
in
Theorem
3.18,
(ii),
when
(g,
r)
=
(0,
3).
On
the
other
hand,
one
verifies
easily,
by
applying
the
theory
of
[CbTpII],
§3,
that,
when
(g,
r)
=
(0,
3),
these
two
definitions
are
in
fact
equivalent.
Indeed,
when
n
=
2
(respec-
tively,
n
≥
3),
this
follows
immediately
from
[CbTpII],
Lemma
3.15,
(ii)
(respectively,
[CbTpII],
Theorems
3.16,
(v);
3.18,
(ii)).
Theorem
3.19
(Metric-,
graph-admissible
outomorphisms
and
tempered
fundamental
groups).
In
the
notation
of
Theorem
3.17,
∧
write
K
for
the
p-adic
completion
of
K;
∧
π
1
temp
((X
K
)
log
n
×
K
K
)
for
the
tempered
fundamental
group
[cf.
[André],
§4,
as
well
as
the
discussion
of
Definition
3.1,
(ii),
of
the
present
paper]
of
(X
K
)
log
n
×
K
∧
K
;
∧
def
=
lim
π
1
temp
((X
K
)
log
Π
tp
n
n
×
K
K
)/N
←−
N
∧
for
the
Σ-tempered
fundamental
group
of
(X
K
)
log
[cf.
n
×
K
K
[CmbGC],
Corollary
2.10,
(iii)],
i.e.,
the
inverse
limit
given
by
allow-
∧
ing
N
to
vary
over
the
open
normal
subgroups
of
π
1
temp
((X
K
)
log
n
×
K
K
)
such
that
the
quotient
by
N
corresponds
to
a
topological
covering
[cf.
[André],
§4.2,
as
well
as
the
discussion
of
Definition
3.1,
(ii),
of
the
present
paper]
of
some
finite
log
étale
Galois
covering
of
∧
(X
K
)
log
of
degree
a
product
of
primes
∈
Σ.
[Here,
we
recall
n
×
K
K
COMBINATORIAL
ANABELIAN
TOPICS
III
95
that,
when
n
=
1,
such
a
“topological
covering”
corresponds
to
a
“com-
binatorial
covering”,
i.e.,
a
covering
determined
by
a
covering
of
the
dual
semi-graph
of
the
special
fiber
of
the
stable
model
of
some
finite
∧
log
étale
covering
of
(X
K
)
log
n
×
K
K
.]
Then
the
following
hold:
(i)
Let
l
∈
Σ
be
such
that
l
=
p.
Then
the
natural
inclusion
Out
{l}-G
(Π
n
)
→
Out(Π
n
)
[cf.
Definition
3.13,
(iv)]
factors
as
a
composite
of
homomor-
phisms
Out
{l}-G
(Π
n
)
−→
Out(Π
tp
n
)
−→
Out(Π
n
)
—
where
the
second
arrow
is
the
natural
homomorphism
[cf.
Proposition
3.3,
(i)].
In
particular,
the
image
of
the
natural
homomorphism
Out(Π
tp
n
)
→
Out(Π
n
)
contains
the
subgroup
{l}-G
Out
(Π
n
)
⊆
Out(Π
n
),
hence
also
the
subgroup
Out
G
(Π
n
)
⊆
Out(Π
n
)
[cf.
Definition
3.13,
(iv)].
(ii)
Write
M
tp
Out
FC
(Π
tp
n
)
⊆
Out(Π
n
)
for
the
inverse
image
of
Out
FC
(Π
n
)
M
⊆
Out(Π
n
)
[cf.
Defi-
nition
3.7,
(iii)]
via
the
natural
homomorphism
Out(Π
tp
n
)
→
Out(Π
n
)
[cf.
(i)].
Then
the
resulting
natural
homomorphism
FC
M
M
Out
FC
(Π
tp
n
)
−→
Out
(Π
n
)
is
split
surjective,
i.e.,
there
exists
a
homomorphism
M
Φ
:
Out
FC
(Π
n
)
M
−→
Out
FC
(Π
tp
n
)
such
that
the
composite
Φ
FC
M
M
Out
FC
(Π
n
)
M
−→
Out
FC
(Π
tp
n
)
−→
Out
(Π
n
)
is
the
identity
automorphism
of
Out
FC
(Π
n
)
M
.
Proof.
Assertion
(i)
follows
immediately
from
Proposition
3.16,
(ii).
Assertion
(ii)
follows
immediately
from
assertion
(i),
together
with
the
fact
that
Out
FC
(Π
n
)
M
⊆
Out
{l}-G
(Π
n
)
[cf.
Theorem
3.17,
(ii)].
This
completes
the
proof
of
Theorem
3.19.
Remark
3.19.1.
In
the
fourth
line
of
the
proof
of
[André],
Proposition
8.6.2,
it
is
asserted
that
one
has
an
injection
alg
Aut
(Γ
alg
0,r+1
)
→
Aut
(Γ
0,r
)
.
In
the
notation
of
the
present
series
of
papers
[cf.
[CmbCsp],
Propo-
sition
1.3,
(vi),
(vii)],
this
homomorphism
corresponds
to
the
natural
homomorphism
Aut
FC
(Π
n+1
)
cusp
−→
Aut
FC
(Π
n
)
cusp
96
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
in
the
case
where
(g,
r,
Σ)
=
(0,
3,
Primes),
and
we
observe
that
Π
n
for
n
≥
1
corresponds
to
“Γ
alg
0,r
for
r−3”
in
the
notation
of
[André],
Proposi-
alg
tion
8.6.2.
However,
this
assertion
is
false.
Indeed,
since
Γ
alg
0,r+1
and
Γ
0,r
are
center-free
[cf.,
e.g.,
[MzTa],
Proposition
2.2,
(ii)],
it
follows
that
the
respective
subgroups
of
inner
automorphisms
determine
compati-
alg
alg
alg
ble
injections
Γ
alg
0,r+1
→
Aut
(Γ
0,r+1
),
Γ
0,r
→
Aut
(Γ
0,r
).
On
the
other
alg
hand,
since
the
natural
surjection
Γ
alg
0,r+1
Γ
0,r
is
far
from
injective,
it
alg
thus
follows
that
the
natural
homomorphism
Aut
(Γ
alg
0,r+1
)
→
Aut
(Γ
0,r
)
also
fails
to
be
injective.
In
particular,
the
proof
given
in
[André]
of
the
injectivity
of
the
first
displayed
homomorphism
GT
(r+1)
−→
GT
(r)
p
p
of
[André],
Proposition
8.6.2,
(1)
—
hence
also
of
•
[André],
Proposition
8.6.2,
(2),
•
[André],
Corollary
8.6.4,
•
the
final
portion
of
[André],
Theorem
8.7.1,
and
•
the
portion
of
[André],
Corollary
8.7.2,
concerning
“GT
(r)
p
”
—
must
be
considered
incomplete.
Moreover,
although
it
is
not
directly
related
to
the
injectivity
of
the
above
discussion,
we
observe
in
passing
[cf.
Remark
3.19.4
below
for
more
details]
that
the
discussion
of
[André],
§8,
also
contains
another
misleading
error.
Remark
3.19.2.
Recall
that,
relative
to
the
notation
of
the
present
series
of
papers,
the
usual
Grothendieck-Teichmüller
group
corresponds
to
the
group
def
GT
=
Out
F
(Π
n
)
Δ+
=
Out
FC
(Π
n
)
Δ+
discussed
in
Theorem
3.18,
(ii)
[cf.
also
Remark
3.18.1],
in
the
case
where
(g,
r,
Σ)
=
(0,
3,
Primes)
[cf.
[CmbCsp],
Remark
1.11.1].
Thus,
from
the
point
of
view
of
the
present
paper,
it
seems
that
one
nat-
ural
candidate
for
the
notion
of
a
local
version
of
the
Grothendieck-
Teichmüller
group
is
the
“metrized
Grothendieck-Teichmüller
group”
def
GT
M
=
Out
F
(Π
n
)
MΔ+
=
Out
FC
(Π
n
)
MΔ+
⊆
GT
discussed
in
Theorem
3.18,
(ii),
again
in
the
case
where
(g,
r,
Σ)
=
(0,
3,
Primes).
Here,
we
recall
that
each
of
these
groups
GT
M
,
GT
admits
a
natural
profinite
topology,
hence,
in
particular,
is
compact
[cf.
Theorem
3.17,
(iv)],
and,
moreover,
is
independent,
up
to
canonical
isomorphism,
of
the
choice
of
n
≥
1
[cf.
Theorem
3.18,
(ii)].
Finally,
one
verifies
immediately
from
the
existence
of
the
natural
splitting
of
the
split
surjection
discussed
in
Theorem
3.19,
(ii)
[cf.
also
the
discussion
COMBINATORIAL
ANABELIAN
TOPICS
III
97
of
the
construction
of
this
splitting
in
the
proof
of
Proposition
3.16,
(ii);
Remark
3.19.3
below]
that,
for
any
positive
integer
n,
one
has
a
natural
inclusion
GT
M
→
GT
(n+3)
p
[cf.
[André],
Notation
8.6.1],
hence
also
a
natural
inclusion
GT
M
→
GT
p
[cf.
[André],
Definition
8.6.3].
In
particular,
one
obtains
a
natural
outer
action
of
GT
M
on
the
“tower”
of
tempered
fundamental
groups
“(Γ
temp
0,r
)
r≥4
”
discussed
in
[André],
Corollary
8.6.4,
i.e.,
in
the
nota-
tion
of
Theorem
3.19
of
the
present
paper,
on
the
system
of
tempered
fundamental
groups
{Π
tp
n
}
n≥1
that
is
manifestly
compatible
with
the
temp
tp
quotients
Π
n
Γ
0,n+3
[cf.
[André],
§8.5].
Remark
3.19.3.
The
construction
of
the
splitting
Φ
given
in
the
proof
of
Theorem
3.19,
(ii),
appears,
at
first
glance,
to
depend
on
the
choice
of
the
prime
l,
as
well
as
on
the
ordering
of
the
n
factors
of
the
con-
figuration
spaces
that
give
rise
to
Π
n
,
Π
tp
n
.
In
fact,
however,
it
is
not
difficult
to
verify
—
by
•
observing
that
symmetries
[e.g.,
that
arise
from
permutations
of
the
n
factors]
of
finite
étale
coverings
of
the
various
configu-
ration
spaces
over
fields
that
appear
always
extend
to
symme-
tries
of
the
corresponding
stable
polycurves
[cf.
the
discussion
of
Remark
3.10.1,
(i),
(e);
[ExtFam],
Theorem
A];
•
applying
the
functoriality
of
the
various
constructions
involved
[cf.
the
discussion
of
“functorial
bijections”
in
the
proof
of
Proposition
3.6]
to
relate
the
“decomposition
groups”
of
the
various
strata
that
appear
in
the
proof
of
Proposition
3.16,
(ii);
and
•
observing
that
these
strata
may
be
described
in
terms
of
“jumps”
in
the
rank
of
the
group-characteristic
sheaf
[cf.
[MzTa],
Def-
inition
5.1,
(i)]
associated
to
the
log
structure
of
the
stable
polycurves
that
appear
[cf.
the
discussion
of
Remark
3.10.1,
(i),
(e)],
hence
are
independent
of
the
ordering
of
the
n
factors
of
the
configuration
spaces
that
appear
—
that
Φ
is
independent
of
the
choice
of
l,
as
well
as
of
the
ordering
of
the
n
factors
of
the
configuration
spaces
that
give
rise
to
Π
n
,
Π
tp
n
.
Remark
3.19.4.
In
passing,
we
observe,
relative
to
the
discussion
of
[André],
§8,
that
the
second
isomorphism
Out
π
1
top
((P
1
C
)
an
\
{x
1
,
.
.
.
,
x
r
})
∼
=
Ker[OutF
r−1
→
GL
r−1
(Z)]
98
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
of
the
final
display
of
[André],
§8.2,
is
false
as
stated
and
should
be
replaced
by
an
inclusion
arrow
“→”.
Indeed,
let
us
first
observe
that
the
first
isomorphism
of
the
final
display
of
[André],
§8.2,
is
correct
as
stated
—
and
indeed
is
a
special
case
of
the
well-known
theorem
of
Dehn-Nielsen-Baer
—
if
one
interprets
the
phrase
“local
monodromies”
in
the
definition
of
“Out
”
as
referring
to
generators
γ
i
of
the
inertia
def
groups
at
the
points
x
i
.
Write
Π
=
π
1
top
((P
1
C
)
an
\
{x
1
,
.
.
.
,
x
r
}).
Then
the
falsity
of
the
second
isomorphism
—
i.e.,
the
non-surjectivity
of
∼
the
natural
inclusion
“→”
induced
by
an
isomorphism
Π
→
F
r−1
—
may
be
verified
as
follows.
First,
we
observe
that
if
one
assumes
the
surjectivity
of
this
natural
inclusion,
then
it
follows
that
the
subgroup
Out
(Π)
⊆
Out(Π)
is
normal.
Thus,
it
suffices
to
obtain
a
contradiction
under
the
assumption
that
the
subgroup
Out
(Π)
⊆
Out(Π)
is
normal.
Next,
let
us
observe
that
the
discrete
free
group
Π
on
r
−
1
generators
is
generated
by
γ
1
,
.
.
.
,
γ
r−1
.
In
particular,
for
any
element
δ
∈
Π
that
appears
as
one
of
a
collection
of
r
−
1
generators
of
Π,
there
exists
an
element
φ
∈
Out(Π)
such
that
φ(δ)
=
γ
1
.
Thus,
since
Out
(Π)
⊆
Out(Π)
is
normal,
it
follows
that
any
element
of
Out
(Π)
preserves
the
def
def
conjugacy
class
of
δ.
Write
δ
1
=
γ
1
·γ
2
;
for
i
=
2,
.
.
.
,
r−3,
δ
i
=
δ
i−1
·γ
i
.
Then,
by
taking
“δ”
to
be
δ
1
,
.
.
.
,
δ
r−3
,
we
conclude
that
any
element
of
Out
(Π)
preserves
the
conjugacy
classes
of
each
of
δ
1
,
.
.
.
,
δ
r−3
.
On
the
other
hand,
one
verifies
immediately
that,
for
a
standard
choice
of
generators
γ
1
,
.
.
.
,
γ
r−1
,
the
elements
δ
1
,
.
.
.
,
δ
r−3
may
be
regarded
as
generators
of
the
nodal
inertia
groups
associated
to
the
nodes
that
appear
in
a
totally
degenerate
pointed
stable
curve
[over
the
field
of
complex
numbers]
that
arises
as
a
degeneration
of
the
pointed
stable
curve
corresponding
to
the
given
Riemann
surface
(P
1
C
)
an
.
Thus,
it
follows
from
[CbTpIV],
Corollary
2.19,
(i);
[CmbGC],
Theorem
1.6,
(ii);
[CmbGC],
Proposition
1.3,
that
any
element
of
Out
(Π)
is
graphic,
i.e.,
in
particular,
preserves
the
conjugacy
classes
of
the
verticial
and
nodal
subgroups
of
Π
that
arise
from
this
totally
degenerate
structure.
On
the
other
hand,
in
light
of
[CbTpIV],
Corollary
2.21,
(iii);
[CbTpIV],
Theorem
2.24,
(ii),
this
implies
that
Out
(Π)
is
an
extension
of
a
finite
group
by
an
abelian
group,
i.e.,
in
contradiction
to
the
fact
that
Out
(Π)
admits
a
surjection
to
a
[highly
nonabelian!]
discrete
free
group
of
rank
≥
2.
Remark
3.19.5.
Finally,
in
passing,
we
note
the
following
conse-
quence,
in
the
context
of
[Tsjm],
of
the
theory
developed
in
the
present
paper.
Suppose,
in
the
notation
of
Theorem
3.18,
(i),
that
n
≥
4
or
r
>
0.
Then
the
homomorphism
Out
FC
(Π
n
)
M
→
Out(Π
tpd
)
M
of
COMBINATORIAL
ANABELIAN
TOPICS
III
99
Theorem
3.18,
(i),
determines
[cf.
[CbTpII],
Definition
3.19]
a
homo-
morphism
Out
F
(Π
n
)
M
→
Out
FC
(Π
tpd
)
MΔ+
=
GT
M
[cf.
the
notation
of
Definition
3.7,
(iii);
Theorem
3.18,
(ii);
Remark
3.19.2].
In
particular,
by
composing
this
last
homomorphism
with
the
restric-
tion
to
GT
M
of
the
homomorphism
of
[Tsjm],
Corollary
B
[cf.
also
[Tsjm],
Remark
2.1.2],
we
obtain
a
natural
composite
homomorphism
Out
F
(Π
n
)
M
→
GT
M
→
G
Q
p
,
whose
restriction,
via
the
natural
homomorphism
I
K
→
Out
F
(Π
n
)
M
[cf.
the
homomorphism
“ρ
n
”
of
Theorem
3.17],
to
I
K
coincides
with
the
natural
outer
homomorphism
I
K
→
G
Q
p
that
arises
from
the
natural
inclusion
of
topological
fields
Q
p
→
K.
Corollary
3.20
(Characterization
of
the
local
Galois
groups
in
the
global
Galois
image
associated
to
a
hyperbolic
curve).
Let
F
be
a
number
field,
i.e.,
a
finite
extension
of
the
field
of
rational
numbers;
p
a
nonarchimedean
prime
of
F
;
F
p
an
algebraic
closure
of
the
p-adic
completion
F
p
of
F
;
F
⊆
F
p
the
algebraic
closure
of
F
in
∧
F
p
;
X
F
log
a
smooth
log
curve
over
F
.
Write
F
p
for
the
completion
def
def
def
of
F
p
;
G
p
=
Gal(F
p
/F
p
)
⊆
G
F
=
Gal(F
/F
);
X
F
log
=
X
F
log
×
F
F
;
π
1
(X
F
log
)
for
the
log
fundamental
group
of
X
F
log
[which,
in
the
following,
we
∧
identify
with
the
log
fundamental
groups
of
X
F
log
×
F
F
p
,
X
F
log
×
F
F
p
—
cf.
the
definition
of
F
!];
∧
π
1
temp
(X
F
log
×
F
F
p
)
∧
for
the
tempered
fundamental
group
of
X
F
log
×
F
F
p
[cf.
[André],
§4];
ρ
X
log
:
G
F
−→
Out(π
1
(X
F
log
))
F
for
the
natural
outer
Galois
action
associated
to
X
F
log
;
∧
:
G
p
−→
Out(π
1
temp
(X
F
log
×
F
F
p
))
ρ
temp
X
log
,p
F
for
the
natural
outer
Galois
action
associated
to
X
F
log
×
F
F
p
[cf.
[André],
Proposition
5.1.1];
∧
Out(π
1
(X
F
log
))
M
⊆
(
Out(π
1
temp
(X
F
log
×
F
F
p
))
⊆
)
Out(π
1
(X
F
log
))
for
the
subgroup
of
M-admissible
outomorphisms
of
π
1
(X
F
log
)
[cf.
Def-
inition
3.7,
(i),
(ii);
Proposition
3.6,
(i)].
Then
the
following
hold:
100
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(i)
The
outer
Galois
action
ρ
temp
factors
through
the
subgroup
X
log
,p
F
∧
Out(π
1
(X
F
log
))
M
⊆
Out(π
1
temp
(X
F
log
×
F
F
p
)).
(ii)
We
have
a
natural
commutative
diagram
G
p
−−−→
Out(π
1
(X
F
log
))
M
⏐
⏐
⏐
⏐
ρ
X
log
F
→
Out(π
1
(X
F
log
))
G
F
−−−
—
where
the
vertical
arrows
are
the
natural
inclusions,
the
upper
horizontal
arrow
is
the
homomorphism
arising
from
the
factorization
of
(i),
and
all
arrows
are
injective.
(iii)
The
diagram
of
(ii)
is
cartesian,
i.e.,
if
we
regard
the
various
groups
involved
as
subgroups
of
Out(π
1
(X
F
log
)),
then
we
have
an
equality
G
p
=
G
F
∩
Out(π
1
(X
F
log
))
M
.
Proof.
Assertion
(i)
follows
immediately
from
the
various
definitions
involved.
Assertion
(ii)
follows
immediately
from
the
injectivity
of
the
lower
horizontal
arrow
ρ
X
log
[cf.
[NodNon],
Theorem
C],
together
with
F
the
various
definitions
involved.
Finally,
we
verify
assertion
(iii).
First,
let
us
observe
that
if
the
smooth
log
curve
“X
F
log
”
is
the
smooth
log
curve
associated
to
P
1
F
\
{0,
1,
∞},
then
assertion
(iii)
follows
immedi-
ately
from
[André],
Theorem
7.2.1.
Write
(X
F
)
log
3
for
the
3-rd
log
con-
log
figuration
space
of
X
F
.
Then
it
follows
immediately
from
[NodNon],
Theorem
B,
that
the
group
Out
FC
(π
1
((X
F
)
log
3
))
of
FC-admissible
outo-
log
morphisms
of
the
log
fundamental
group
π
1
((X
F
)
log
3
)
of
(X
F
)
3
[which,
in
the
following,
we
identify
with
the
log
fundamental
groups
of
(X
F
)
log
3
×
F
∧
log
F
p
,
(X
F
)
3
×
F
F
p
—
cf.
the
definition
of
F
!]
may
be
regarded
as
a
closed
subgroup
of
Out(π
1
(X
F
log
)).
Moreover,
it
follows
imme-
diately
from
the
various
definitions
involved
that
the
respective
images
Im(ρ
X
log
),
Im(ρ
temp
)
of
the
natural
outer
Galois
actions
ρ
X
log
,
ρ
temp
X
log
,p
X
log
,p
F
F
F
F
associated
to
X
F
log
,
X
F
log
×
F
F
p
are
contained
in
this
closed
subgroup
log
Out
FC
(π
1
((X
F
)
log
3
))
⊆
Out(π
1
(X
F
)).
Thus,
to
verify
assertion
(iii),
one
verifies
easily
that
it
suffices
to
verify
the
equality
M
Im(ρ
temp
)
=
Im(ρ
X
log
)
∩
Out
FC
(π
1
((X
F
)
log
3
))
X
log
,p
F
F
[cf.
Definition
3.7,
(iii)].
On
the
other
hand,
since
the
“ρ
X
log
”
that
F
occurs
in
the
case
where
we
take
“X
F
log
”
to
be
the
smooth
log
curve
associated
to
P
1
F
\
{0,
1,
∞}
is
injective
[cf.
assertion
(ii)],
this
equality
COMBINATORIAL
ANABELIAN
TOPICS
III
101
follows
immediately
—
by
considering
the
images
of
the
subgroups
M
Im(ρ
temp
)
⊆
Im(ρ
X
log
)
∩
Out
FC
(π
1
((X
F
)
log
3
))
X
log
,p
F
F
M
of
Out
FC
(π
1
((X
F
)
log
via
the
tripod
homomorphism
associated
to
3
))
log
FC
Out
(π
1
((X
F
)
3
))
[cf.
[CbTpII],
Definition
3.19]
—
from
Theorem
3.18,
(i),
together
with
assertion
(iii)
in
the
case
where
we
take
“X
F
log
”
to
be
the
smooth
log
curve
associated
to
P
1
F
\
{0,
1,
∞}
[which
was
ver-
ified
above].
This
completes
the
proof
of
assertion
(iii),
hence
also
of
Corollary
3.20.
Remark
3.20.1.
Corollary
3.20,
(iii),
may
be
regarded
as
a
gen-
eralization
of
[André],
Theorems
7.2.1,
7.2.3,
obtained
at
the
cost
of
replacing,
in
effect,
Out(π
1
(X
F
log
))
G
by
the
possibly
smaller
group
Out(π
1
(X
F
log
))
M
⊆
Out(π
1
(X
F
log
)).
Here,
we
note
that
unlike
the
sub-
groups
G
p
⊆
G
F
[cf.,
e.g.,
[AbsHyp],
Theorem
1.1.1,
(i)]
and
Out(π
1
temp
∧
∼
(X
F
log
×
F
F
p
))
→
Out(π
1
(X
F
log
))
G
⊆
Out(π
1
(X
F
log
))
[cf.
Definition
3.7,
(i);
Proposition
3.6,
(i);
Remark
3.13.1,
(i);
Theorem
3.17,
(v)],
which
are
commensurably
terminal,
the
subgroup
Out(−)
M
⊆
Out(−)
fails,
in
general
[at
least
in
the
pro-l
case],
even
to
be
normally
terminal
[cf.
Remark
3.17.1].
Remark
3.20.2.
Let
us
recall
that,
in
the
proof
of
[NodNon],
Theorem
C,
the
authors
applied
•
the
injectivity
portion
of
the
theory
of
combinatorial
cuspidal-
ization,
together
with
•
the
injectivity
of
the
outer
Galois
representation
associated
to
a
tripod,
to
prove
•
the
injectivity
of
the
outer
Galois
representation
associated
to
an
arbitrary
hyperbolic
curve.
On
the
other
hand,
in
the
proof
of
Corollary
3.20,
the
authors
applied
•
the
[almost
pro-l]
injectivity
portion
of
the
theory
of
combinato-
rial
cuspidalization
[in
the
form
of
Theorem
3.18,
(i)],
together
with
•
the
characterization
of
the
local
Galois
groups
in
the
global
Galois
image
for
tripods,
to
prove
•
an
analogous
characterization
of
the
local
Galois
groups
in
the
global
Galois
image
for
arbitrary
hyperbolic
curves.
102
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
The
formal
similarity
of
these
two
proofs
suggests
that
it
is
perhaps
natural
to
think
of
the
injectivity
portion
of
the
theory
of
combina-
torial
cuspidalization
as
a
sort
of
tool
for
reducing
certain
problems
concerning
arbitrary
hyperbolic
curves
to
the
case
of
tripods.
Remark
3.20.3.
By
comparison
to
André’s
original
characterization
of
the
local
Galois
groups
in
the
global
Galois
image
[cf.
[André],
Theorems
7.2.1,
7.2.3],
from
the
point
of
view
of
a
researcher
who
is
interested
only
in
tripods
[i.e.,
not
in
arbitrary
hyperbolic
curves],
the
motivation
for
the
theory
developed
in
the
present
paper
concerning
Out(−)
M
may
at
first
glance
appear
insufficient.
In
fact,
however,
as
discussed
in
Remarks
3.19.1,
3.19.2,
even
if
one
is
interested
only
in
tripods,
it
is
necessary
to
apply
the
extensive
theory
developed
in
the
present
paper
concerning
Out(−)
M
in
order
to
repair
the
mistake
in
[André]
and
realize
the
original
goal
of
the
present
paper,
i.e.,
of
defin-
ing
a
suitable
local
analogue
of
the
Grothendieck-Teichmüller
group.
COMBINATORIAL
ANABELIAN
TOPICS
III
103
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